Home » Mathematics » Pre Calculus » Probability

Vincent Selhorst-Jones

Vincent Selhorst-Jones

Probability

Slide Duration:

Table of Contents

Section 1: Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Probability

  • In the lesson on Counting, we defined the term event to simply mean something happening. For example, if we flipped a coin, we might name an event E that is the event of the coin coming up heads. Of course, there might be possibilities other than the event occurring. We call the set of all possible outcomes the sample space. If we want, we can denote the sample space with a symbol, such as S. In the coin example, there are two possible outcomes in the sample space S: heads and tails.
  • Let E be an event and S be the corresponding sample space. Let n(E) denote the number of ways E can occur and n(S) denote the total possible number of outcomes. Then if all the possible outcomes in S are equally likely, the probability of event E occurring (denoted P(E)) is
    P(E) =n(E)
    n(S)


     
    		.
    Equivalently, using words,
    Probability of event =# of ways event can occur
    # of possible outcomes


     
    		.
  • Notice that in the previous definition, it was assumed that all the possible outcomes were "equally likely". If this isn't true, the above does not work. Happily, we are almost certainly not going to see any problems that don't involve equal likelihood of all the outcomes. It might be described in the problem as `fair', `random', or something else, but we can almost always assume that all possible outcomes are equally likely at this level in math.
  • We can represent probability as a fraction ([1/2]), a decimal (0.5), or a percentage (50%). In any case, a probability is always between 0 and 1, inclusive. The larger the value, the more likely. We can also interpret probability as the ratio of the event happening over a large number of attempts. For example, if we flip a coin a million times, we can expect about half of the flips to come out heads. (P(Eheads) = [1/2])
  • Given two mutually exclusive events A and B, the probability of either one (or both) occurring (A∪B) is given by
    P(A ∪B) = P(A) + P(B) .
    If the events are not mutually exclusive, we have to take the overlap into account. We can represent where A and B overlap with A ∩B. Then
    P(A∪B)  = P(A) + P(B) − P(A∩B).
  • If we have some event E, we can talk about the event of E not occurring. We call this the complement of E, denoting it as Ec. [Other textbooks/teachers might denote it ―E or E′.] The probability of an event's complement occurring is
    P(Ec) = 1 − P(E).
  • Two events are independent if they are separate events and the outcome of either one does not affect the other. Given two independent events A and B, the probability of both events occurring is
    P(A and B)  = P(A) ·P(B).
  • If the events are not independent (the outcome of one does affect the other) and we want to find the chance of them both occurring, we need the idea of conditional probability. We denote the conditional probability of B occurring if A does occur as P(B | A). (We can interpret this as the probability of B happening if we are guaranteed that A will happen.) Then, given two events A and B, where the outcome of A affects the outcome of B, the probability of both events occurring is given by
    P(A and B)  = P(A) ·P(B | A).

Probability

Given a fair, six-sided die, what is the chance of rolling a 1, 5, or 6 on it?
  • A fair die is one where each side coming up is equally likely. When all the ways that something can occur are equally likely, the probability of something happening is quite simple:
    Probability of event =# of ways event can occur
    # of possible outcomes
  • For this problem, there are three different ways the event we are interested in can occur: we can roll a 1, we can roll a 5, or we can roll a 6.
    # of ways event can occur     =     3
    Since the die has six sides, there are a total of 6 different possible outcomes when we roll the die.
    # of possible outcomes     =     6
  • To find the probability of the event occurring, simply divide the number of ways the event we are interested in can happen by the total number of possible outcomes:
    P     =     3
    6
[1/2]
Given a standard deck of 52 playing cards, what is the probability of drawing a card at random from the deck and winding up with a 7, 8, or 9?
  • If you don't know how a standard deck of 52 playing cards is made up, start off by doing a little research. You need to know what you're working with before you can solve a problem. Looking it up, we find that a standard deck of cards has four suits (each with one of two colors): `Spades' (black), `Hearts' (red), `Clubs' (black), `Diamonds' (red). In each of these suits, there are 13 different cards. The cards are `Ace', the numbers 2-10, `Jack', `Queen', and `King'. Finally, the last three cards (J, Q, K) are sometimes called `Face' cards because they have a picture of a person on them.
  • Since we're told that we draw the card randomly from the deck, we know that we are equally likely to pull out any card, so we have
    Probability of event =# of ways event can occur
    # of possible outcomes
  • Counting up the number of 7's, 8's, and 9's in the deck, we see that they will appear once in every suit. Since there are four suits, that means we have 3·4=12 of them in the deck. The deck has a total of 52 cards, so the probability is
    P     =     12
    52
    		    =     3
    13
    [Notice that we could also get directly to the simplified fraction of [3/13] by the following logic: when we pull a card from the deck, it must be one of the suits. Thus we only need to know the chance of getting a 7, 8, or 9 within a single suit. Since there are 13 cards in each suit, we have [3/13].]
[3/13]
A group of eight people is going to be randomly seated around a circular table, where all the seats are evenly arranged. Within that group of eight people are two friends. What is the chance that the two friends will be seated next to one another?
  • Begin by visualizing the problem. There is a round table with eight chairs sitting around it. If you have difficulty understanding it, you might find it useful to draw a quick sketch: perhaps a circle with eight little boxes around it. Eight people will randomly sit down, and we're interested in the chance of two specific people sitting next to each other.
  • We can approach the problem like this: one of the two friends will be seated somewhere. We don't know where, but the first friend will have to be seated at one of the eight seats-this is guaranteed. From there, how many ways can the second friend be seated next to the first friend? How many ways could the second friend be seated total?
  • Remember, since it's a circular table, the second friend could be seated to the right or to the left of the friend. Thus there are 2 possible ways they could be seated next to each other. After the first friend has been seated, there are only seven seats left at the table (8−1), so there are a total of 7 possible ways for the second friend to be seated. Since the seating is random, we have a probability of [2/7].
[2/7]
A fair, six-sided die is rolled three times in succession. What is the probability that the first roll will be a 4, 5, or 6, the second roll will be a 5 or 6, and the third roll will be a 6?
  • Begin by noticing that each roll of the die is unaffected by the other rolls. The probability of the first roll coming up 4, 5, or 6 is unaffected by what happens in the other rolls, and similarly for the second and third rolls being unaffected by the others. This means that we have independent events: we can find the probability of all the events occurring by multiplying each of the sub-events together.
  • The probability of the first roll coming up 4, 5, or 6 is [3/6] (3 ways it can happen, 6 possible outcomes). The probability for the second roll coming up 5 or 6 is [2/6] (2 ways it can happen, 6 possible outcomes). The probability for the third roll coming up 6 is [1/6] (1 way it can happen, 6 possible outcomes).
    Pfirst = 3
    6
    		 ,        Psecond = 2
    6
    		 ,        Pthird = 1
    6
  • Since each of the three rolls must occur for the event we are interested in (the event is the three rolls each coming out as specified in the problem), we can multiply them all together to find the probability of the event:
    P     =     Pfirst ·Psecond ·Pthird     =     3
    6
    		· 2
    6
    		· 1
    6
    		    =     1
    2
    		· 1
    3
    		· 1
    6
    		    =     1
    36
[1/36]
At a certain school, on any given day the probability that a random student does not eat anything for breakfast is 23%. If the school has a total of 1200 students, how many students eat breakfast on any given day?
  • Begin by noticing that `not eating anything for breakfast' and `eating breakfast' are opposite conditions. If one occurs, the other must not, and vice-versa. In "math-speak", these are called complementary events: if something happens, it must be in one of the events, and it cannot be in both of them.
  • The probability of an event's complement (its "opposite") occurring is
    1−P,
    where P is the probability of the event occurring. For this problem, we were given the probability as P = 23%, so we must first convert it to decimal form: P = 0.23. Once we have that, we can find the probability of the complement:
    1− 0.23     =     0.77
    [Notice that we could also do this with percentages: 100% − 23% = 77%. Converting to decimal format is convenient for the next step, though.]
  • Now that we know the probability of a student eating breakfast is 0.77, we can multiply that by the total number of students to find what portion of them will fulfill that event-that is, eat breakfast:
    0.77·1200     =     924
924
You have lots of fair, six-sided dice in front of you. What is the minimum number of dice you must roll at the same time so that the probability of at least one of them coming up with a 5 is at least 50%?
  • Begin by understanding the question. You can roll some number of dice (you choose how many you roll), then you look at the results of those dice. Your job is to figure out how many dice need to be rolled for the chance of at least one die coming up as a 5 to be greater than or equal to 50%.
  • Before we solve the problem, let's look at a common mistake people make when attempting to solve this kind of problem: lots of people assume that since the chance of one die coming up as a 5 is [1/6], if we roll six dice, we will be guaranteed a 5. Since that is 100%, half as many dice would produce 50%, so three dice is the required number. The above logic is faulty for a couple of reasons, but the simplest one is that you can never guarantee a certain number will be rolled on a fair die. If you roll a hundred dice, there is always the chance that they could all come out as 1's, or anything else. The chance may be tiny at that point, but it is impossible to be completely certain. That 100% probability the "logic" was built on is impossible to achieve.
  • Instead, a good way is to figure out the chance of a 5 not being rolled. Once we know the chance of that happening, we can figure out the complement: the chance of one or more 5's being rolled. For example, at one die, the chance of a 5 not being rolled is [5/6]. Thus the chance of a 5 coming up is
    One die:        1 − 5
    6
    		    =     1
    6
  • Work through this with increasing number of dice until you find the first one that a probability of 50% occurs at. For two dice, the chance of a 5 not being rolled on either die is [5/6] ·[5/6], so the probability of a 5 coming up somewhere on two dice is
    Two dice:        1 −⎛
    ⎝    		 5
    6
    		· 5
    6
    		⎞
    ⎠    		   ≈     0.306
    For three dice, the chance of a 5 not coming up is [5/6] ·[5/6] ·[5/6], so the probability of a 5 coming up somewhere on three dice is
    Three dice:        1 −⎛
    ⎝    		 5
    6
    		· 5
    6
    		· 5
    6
    		⎞
    ⎠    		   ≈     0.421
    For four dice, the chance of a 5 not coming up is [5/6] ·[5/6] ·[5/6] ·[5/6], so the probability of a 5 coming up somewhere on four dice is
    Four dice:        1 −⎛
    ⎝    		 5
    6
    		· 5
    6
    		· 5
    6
    		· 5
    6
    		⎞
    ⎠    		   ≈     0.518
    Thus the minimum number of dice to guarantee a 50% or better chance is four dice.
4
Given a standard deck of 52 playing cards, what is the chance of randomly drawing three cards and them all being face cards?
  • If you don't know how a standard deck of 52 playing cards is made up, start off by doing a little research. You need to know what you're working with before you can solve a problem. Looking it up, we find that a standard deck of cards has four suits (each with one of two colors): `Spades' (black), `Hearts' (red), `Clubs' (black), `Diamonds' (red). In each of these suits, there are 13 different cards. The cards are `Ace', the numbers 2-10, `Jack', `Queen', and `King'. Finally, the last three cards (J, Q, K) are sometimes called `Face' cards because they have a picture of a person on them. Since there are three face cards for each suit, and there are four suits in a deck, a 52 card deck begins with 3·4 = 12 face cards.
  • Notice that, unlike previous problems, the card that gets pulled first affects the probability of later cards being pulled. If a face card is pulled, there is less chance of pulling a face card later. If a face card is not pulled, there is more of a chance of pulling a face card later. To deal with the fact that each pull affects the subsequent ones, we can use the idea of conditional probability. If we know the chance of a first event occurring, and we know the chance of a second event occurring assuming the first event occurs, we can multiply them together to find the chance of both events occurring. For this specific problem, that means we can find the chance of the first pull being a face, then the chance of the second (assuming the first pull was a face), then the chance of the third (assuming the first two pulls were faces), then finally multiply them all together.
  • Since a 52 card deck has 12 face cards to begin with, the chance of the first face being pulled is
    First face:        12
    52
    Now we move on to the next card. At this point, the deck has one less card and one less face, so there are 51 cards with 11 faces shuffled in:
    Second face:        11
    51
    Finally we have the third card. At this point, the deck is two cards down and two faces down, so there are 50 cards with 10 faces:
    Third face:        10
    50
    To find the probability of the event occurring where each pull happens, we multiply them all together:
    P     =     12
    52
    		· 11
    51
    		· 10
    50
    		    =     11
    1105
    		   ≈     0.009  955
[12/52] ·[11/51] ·[10/50]     =     [11/1105]    ≈     0.009  955
In a sack of marbles, there are 8 green marbles, 12 blue marbles, 15 red marbles, and 5 yellow marbles for a total of 40 marbles in the sack. If you reach into the sack and draw out three marbles at random, what is the chance that you will draw precisely 1 green marble and 2 red marbles?
  • Notice that drawing a given marble affects the chances of drawing other marbles. This means that if we want to approach this problem in a step-by-step manner (first marble, second marble, third marble), we need to use conditional probability. However, unlike the previous problem, there are multiple ways for the event we are interested in to occur. The event could happen in a variety of different orders, shown below (where G means a green marble is pulled and R means a red marble is pulled):
    Possible orders:       GRR,       RGR,        RRG
  • Since the event can occur using any of these three different ways, we will sum the probabilities for each of the orders to find the total probability of the event. Working through each order with conditional probability, we have
    GRR        ⇒        8
    40
    		· 15
    39
    		· 14
    38

    RGR        ⇒        15
    40
    		· 8
    39
    		· 14
    38

    RRG        ⇒        15
    40
    		· 14
    39
    		· 8
    38
  • Thus, since the event can happen by following any of these three orders, we add them all together to find the probability of the event happening by any of the orders:
    ⎛
    ⎝    		 8
    40
    		· 15
    39
    		· 14
    38
    		⎞
    ⎠    		 +⎛
    ⎝    		 15
    40
    		· 8
    39
    		· 14
    38
    		⎞
    ⎠    		 +⎛
    ⎝    		 15
    40
    		· 14
    39
    		· 8
    38
    		⎞
    ⎠
    Simplify:
    15 ·14 ·8
    40 ·39 ·38
    		 + 15 ·14 ·8
    40 ·39 ·38
    		 + 15 ·14 ·8
    40 ·39 ·38
    		    =     3 ·⎛
    ⎝    		 15 ·14 ·8
    40 ·39 ·38
    		⎞
    ⎠    		   ≈     0.085
  • Alternatively, there is another way to do this that does not involve conditional probability and is arguably simpler. Remember, since the drawing is random, we have
    Probability of event =# of ways event can occur
    # of possible outcomes
    Using combinations from the previous lesson, it's easy to see how many total possible outcomes there are: we're choosing three marbles from a bag of 40, so it's 40C3. To figure out the number of ways that 1 green and 2 reds can be pulled, look at the number of ways each marble can be pulled from its group: there are 8C1 ways to pull the green and 15C2 ways to pull the two reds. If we want to know all the ways we can do both, we multiply them because each green still allows the pulling of reds, and each red still allows the pulling of a green. Thus the total number of ways the event can happen is (8C1) ·(15C2). This means the probability of the event is
    (8C1) ·(15C2)
    40C3
    		    =    
    8!
    1!·7!
    		· 15!
    2! ·13!
    40!
    3!·37!
    Simplify:
    8!
    1!·7!
    		· 15!
    2! ·13!
    40!
    3!·37!
    		    =    
    8
    1
    		· 15 ·14
    2
    40·39 ·38
    3 ·2
    		    =     8 ·15 ·7
    40 ·39 ·38
    		·3 ·2     =     3 ·⎛
    ⎝    		 15 ·14 ·8
    40 ·39 ·38
    		⎞
    ⎠    		   ≈     0.085
    Which is exactly what we got doing it with conditional probability, so the two methods check each other. Great!
3 ·( [(15 ·14 ·8)/(40 ·39 ·38)] )    ≈     0.085
At a popular game show, four contestants are randomly selected from the audience. At today's filming for the show, there are 47 students from a nearby college. If the total number of people in the audience is 400, what is the chance that precisely two students from the college will be picked to be contestants?
  • Note: To do this problem, we will use the idea of a combination from the previous lesson. If you are not familiar with this idea, you will need to watch the part of the previous lesson dealing with combinations before the below makes sense.
  • Begin by making sense of the problem: there is an audience comprised of two groups-college and non-college. There are a total of 400 people, with 47 of those being in the college group. From the entire audience of 400, four different people are selected. Our job is to figure out what the probability is of precisely 2 college and 2 non-college being picked together.
  • We might consider doing this problem by creating a lengthy tree diagram, where each branch represents one of the possible events (college or non-college picking). We can calculate the probability of a given path occurring by multiplying together all the probabilities for the branches in that path. Then we can find the total probability of our event (2 college and 2 non-college) occurring by adding up all those paths. This method will work. It is also extremely difficult and time-consuming to do the above. Instead, there is a much easier way. Remember, since the selection is random, the probability of the event happening is quite simple:
    Probability of event =# of ways event can occur
    # of possible outcomes
  • This means all we need to do is compute the number of ways the event can occur and the number of all possible outcomes. To figure out the ways the event can occur, notice that we're interested in 2 college and 2 non-college being picked. Since we have 47 college students total, all the ways that 2 college students can be chosen is 47C2. We can work things out similarly for all the ways 2 non-college can be picked: there are 400−47=353 non-college in the audience, so the number of possible ways is 353C2. Thus, the number of possible ways both things can happen is the product:
    # of ways to pick 2 college and 2 non−college:        (47C2) ·(353C2)
    The total number of possible outcomes is much simpler: it's just all the ways that 4 people can be selected from an audience of 400:
    # of ways to pick 4 people from audience:        400C4
  • To find the probability, we divide the ways the event can happen by all the possible outcomes, then simplify:
    (47C2) ·(353C2)
    400C4
    		    =    
    47!
    2!·45!
    		· 353!
    2!·351!
    400!
    4!·396!
    		    =    
    47·46
    2
    		· 353 ·352
    2
    400 ·399 ·398 ·397
    4 ·3 ·2
    To make it easier to see, we can multiply the numerator by the denominator's reciprocal:
    47 ·46 ·353 ·352
    2 ·2
    		· 4 ·3 ·2
    400 ·399 ·398 ·397
    		    =     47 ·46 ·353 ·352 ·6
    400 ·399 ·398 ·397
    Finally, use a calculator to simplify and get an approximate decimal:
    16 790 092
    262 684 975
    		   ≈     0.0639
[(16 790 092)/(262 684 975)]    ≈     0.0639
If you randomly draw five cards from a standard deck of 52 playing cards, what is the percentage chance that you will draw a full house?
  • Note: To do this problem, we will use the idea of a combination from the previous lesson. If you are not familiar with this idea, you will need to watch the part of the previous lesson dealing with combinations before the below makes sense.
  • If you don't know what a full house is or how a standard deck of 52 playing cards is made up, start off by doing a little research. You need to know what you're working with before you can solve a problem. Looking up full house, we find that it is a set of five cards where 3 of the cards all match each other and the remaining 2 also match. Notice that a full house has nothing to do with the suits of the cards: they merely need to match in terms of being the same number/face/ace. Thus a full house contains one three-of-a-kind and another pair. Some examples: 777KK,  AAA33,  88855, etc. From previous problems, we've learned that a standard deck of cards has four suits. In each of these suits, there are 13 different cards. The cards are `Ace', the numbers 2-10, `Jack', `Queen', and `King'.
  • We might consider doing this problem by creating a lengthy tree diagram, where each branch represents one of the possible events (which card is picked next). We can calculate the probability of a given path occurring by multiplying together all the probabilities for the branches in that path. Then we can find the total probability of our event (full house) occurring by adding up all those paths. This method will work. It is also extremely difficult and time-consuming to do the above. Instead, there is a much easier way. Remember, since the selection is random, the probability of the event happening is quite simple:
    Probability of event =# of ways event can occur
    # of possible outcomes
  • This means all we need to do is compute the number of ways a full house can occur along with the total number of ways five cards can be drawn from 52. First off, notice that the order of the cards is unimportant: you still have a full house whether you draw in the order of 33322 or 2 33 2 3 or anything else. This means we will be using combinations, not permutations. This problem is not about ordering, it is about choosing. To compute the number of ways to draw three-of-a-kind and a pair, let's start with the three-of-a-kind first. We can do this with any of the 13 ranks of card, and once we choose a rank (A,2,3,…,Q,K), we then need to pull three of them out of the four total. Thus, for the three of a kind, there are 13 ·(4C3). For the pair, we can do it with any of the remaining 12 ranks, and we pull out two of them, so 12 ·(4C2). To find all the ways to do both of these, we multiply them together:
    # of ways to draw full house:       ⎡
    ⎣    		 13 ·(4C3)⎤
    ⎦    		·⎡
    ⎣    		 12 ·(4C2)⎤
    ⎦
    To find the total number of ways we can draw five cards, that's much simpler: all the ways that five objects can be chosen from 52.
    # of ways to draw 5 cards:        52C5
  • To find the probability, divide the ways the event can happen by all the possible outcomes:
    13 ·12 ·(4C3) ·(4C2)
    52C5
    		    =    
    13 ·12 · 4!
    3!·1!
    		· 4!
    2!·2!
    52!
    5!·47!
    Simplify a bit:
    13 ·12 ·4 · 24
    4
    52 ·51 ·50 ·49 ·48
    120
    		    =     13·12 ·4 ·6 ·120
    52 ·51 ·50 ·49 ·48
    		   ≈     0.00144
    Finally, the problem asked for a percentage chance, not a decimal, so convert:
    0.00144     =     0.144%
0.144%

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Probability

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:06
  • Definition: Sample Space 1:18
    • Event = Something Happening
    • Sample Space
  • Probability of an Event 2:12
    • Let E Be An Event and S Be The Corresponding Sample Space
  • 'Equally Likely' Is Important 3:52
    • Fair and Random
  • Interpreting Probability 6:34
    • How Can We Interpret This Value?
    • We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
  • One of Multiple Events Occurring 9:52
    • Mutually Exclusive Events
    • What If The Events Are Not Mutually Exclusive?
    • Taking the Possibility of Overlap Into Account
  • An Event Not Occurring 17:14
    • Complement of E
  • Independent Events 19:36
    • Independent
  • Conditional Events 21:28
    • What Is The Events Are Not Independent Though?
    • Conditional Probability
    • Conditional Events, cont.
  • Example 1 25:27
  • Example 2 27:09
  • Example 3 28:57
  • Example 4 30:51
  • Example 5 34:15

Transcription: Probability

Hi--welcome back to Educator.com.0000

Today, we are going to talk about probability.0002

Consider if we wanted to know the chance of rain today, the odds of winning a bet, or how likely a medicine is to cure a disease.0004

In all of these cases, we would be asking the probability of something happening.0010

The study of probability and chance is a major area of mathematics.0014

It has applications throughout business, science, politics, medicine, and many other fields.0018

Being able to know how probable an event is, how likely something is to happen, is extremely important for a huge number of things.0022

In this lesson, we will go over some of the basic concepts of probability.0029

Still, even though they are basic concepts, it is going to be a really interesting amount of stuff that is going to let us see some really interesting results.0032

There is a lot of stuff here, even at the basic level.0039

Furthermore, basic probability questions pop up a lot on standardized tests, like the SAT and other standardized tests.0042

So, if you are planning on taking any of those in the near future, this is especially useful,0049

because you are almost certainly going to see some questions that are exactly like what we are working on here, but probably even easier.0052

And while it is not absolutely necessary to have watched the previous two lessons before watching this one,0057

we are going to draw very heavily on the previous two lessons in this lesson.0066

So, I would really recommend watching those first, if you haven't watched them.0069

It is not absolutely necessary, but it will make things a little clearer.0072

All right, let's go: the first thing that we want to define is the idea of a sample space.0075

In the lesson on counting, we defined the term event to simply mean something happening.0079

For example, if we had a coin, and we flipped it, we might name an event E that is the event of the coin coming up heads.0085

Of course, there might be possibilities other than the event occurring.0092

It might come up something other than heads.0095

We call the set of all possible outcomes, everything that could happen when we do something, the sample space.0097

If we want, we can denote the sample space with a symbol, such as S.0104

In the example above, there are two possible outcomes for the sample space.0108

1) It comes up heads, or 2) it comes up tails.0111

There are two different possibilities, because there are two different sides to the coin.0115

Two sides to the coin means two things in our sample space.0120

The event is simply heads coming up; the sample space is everything that could occur, heads and tails.0123

Probability of an event: let E be an event, and S be the corresponding sample space.0131

Let n(E) denote the number of ways that E can occur, and n(S) denote the total possible number of outcomes.0136

Then, if all of the possible outcomes in S are equally likely, the probability of event E occurring,0143

denoted p(E), probability of E, is p(E) = n(E)/n(S).0149

Equivalently, just using words, that is: the probability of an event is equal to the number of ways the event can occur, divided by the number of possible outcomes.0155

For example, in a standard 52-card deck, there are 4 of each card in various suits.0167

If we draw a card at random from the deck, the chance of drawing an ace is...0172

The probability of the event of an ace coming up is: there are four ways that we can get an ace out of the deck.0177

There are four aces in there, so there are four ways that that can happen.0187

And there are a total of 52 cards in the deck, so there is a total possibility...the total number of possible outcomes0190

is pulling any of those 52; so it is 4/52, which simplifies to 1/13.0195

This idea is the basic idea of probability: being able to say the number of ways it could occur, divided by the number of all possible things that could happen.0201

That is pretty much the main idea; and if you take one idea away from this lesson, this is the one idea to take away.0210

And this is the sort of thing that you will end up seeing on any standardized test.0215

This idea right here is enough for any standardized test.0218

So, as long as you keep that one in your mind, you will be good for all of that.0220

We are going to get into slightly more interesting things as we keep going, but this is the basic, fundamental idea that you want to hold onto.0223

Equally likely is important: that is a really important phrase.0231

In that previous definition, it was assumed that all possible outcomes were equally likely.0234

This is a really important requirement for the way that we are going to look at probability.0243

Why? Well, let's consider the following scenario.0247

If you dig a hole in the ground, there is a possibility that you will find gold.0249

The sample space, then, has two possibilities: you find gold, or you don't find gold.0255

We have an event, which is finding gold; and our sample space is finding gold or not finding gold.0261

That means one thing for the event is divided by two things for the size of our sample space.0267

But clearly, if you dig some random hole, the chance of you finding gold is not 1/2.0272

Why? Because each chance is not equally likely.0278

Each outcome is not equally as likely as the other one.0282

Finding gold is not equal occurrence, equal probability, with not finding gold; they are not equally likely outcomes.0285

So, because they are not equally likely outcomes, we can't base it off of this idea of0292

number of ways of our event, divided by number of possible ways anything could happen.0295

All right, the method that we just talked about for that basic probability thing won't work if we don't have this "equally likely" thing.0301

This "equally likely" thing is a really important first requirement.0309

Now, happily, we are almost certainly never going to see anything that doesn't involve equal likelihood.0314

All of the problems that we are going to end up seeing are going to be equal likelihood problems.0320

We are going to know that all of the outcomes are equally likely.0325

They might describe it with words like fair, like "a fair die" or "a fair coin" or a "fair" random number--0328

something that implies that all of the possibilities are equally likely.0337

A fair die is one that is equally likely to come up 1, 2, 3, 4, 5, 6.0342

All of its outcomes are equally likely.0347

We can also use the word "random"; if something is selected randomly, that is implying that, out of the selection, it was equally likely amongst all of them.0349

Or some other way of saying this...but we can almost always assume that all of the possible outcomes are equally likely.0356

At the level of the problems that we are going to be working at, this is a pretty reasonable assumption to make:0364

that we can assume this "equally likely" thing at the level of the problems we are working at.0368

There are lots of really interesting problems you can work on, where this assumption won't end up being true.0373

But for the sort of thing you will be required to work on at this point,0378

you can almost always be certain that you will be allowed to assume0381

that everything is equally likely, unless they very explicitly tell you otherwise.0384

And that is not going to happen very often, if at all.0387

All right, interpreting probability: notice that the probability of an event, p(E) = n(E)/n(S), is always less than or equal to 1.0393

This is because n(E) ≤ n(S); n(E), the number of ways the event can occur,0401

is always less than or equal to the total number of things that can happen,0407

because all of the ways that the event can occur are all inside of the ways that anything could happen.0410

E is always contained within our sample space; the event is always contained within the sample space,0416

just like heads was contained inside of heads and tails for the sample space.0422

The event is always contained inside of the sample space; so the number of ways the event can happen is always going to be smaller,0427

or equal to, the number of things in the sample space, total.0432

OK, we have this number, then, that can be somewhere between 0 and 1.0436

The smallest E could be is 0 ways total; so we are somewhere between 0 and 1.0440

How do we interpret this value?0444

We interpret it like this, where we have this value that can range between 0 and 1.0446

And at 0, it is absolutely impossible for the thing to occur; and at 1, it is absolutely certain that the thing will occur.0450

It will definitely occur at a 1 probability, and it will never occur at a 0 probability.0459

So, as we end up going up the scale, as we go up from 0 to 1, it becomes more and more likely.0463

The closer you get to 1, the more likely it is.0470

And here in the middle, at 0.5, it is equally likely as unlikely.0472

On average, 1 out of 2 times it will end up happening.0476

We can represent probability as a fraction, as a decimal, and as a percentage.0481

Any of these are fine things to do; the important part is that probability is always between 0 and 1, inclusive,0487

because you can be 0, and you can be 1, as a probability,0494

although almost all of the ones we are going to deal with will be somewhere in between.0497

We can also interpret probability as the ratio of the event happening over a large number of attempts.0500

For example, if we flip a coin a million times, we can expect about half of the flips to come out heads,0506

because the probability of flipping a fair coin and having it come out heads is 1 out of 2, 1/2.0512

On the large scale, we know that about half of any large thing will end up coming out to be that.0518

Now, on the small scale, if I flip a coin twice, it wouldn't be that surprising for two of them to come out as tails,0523

even though it is a 1-out-of-2 chance for heads.0529

We could flip the coin three times, and it wouldn't be that surprising for it to come out as tails, tails, tails.0532

It is not super likely, but it is not that unreasonable.0537

Just because the probability is 1/2 for heads doesn't guarantee us that it is going to occur any time.0540

With probability, we don't have a guarantee of occurrence; we just have "likely" that it will occur at certain levels of likelihood.0546

We can only have certainty at a 1.0554

So, since it is a question of how likely it is, we won't have it be as likely to show up, unless we look at a larger sample space.0556

We look at a larger number of things that could happen.0565

If we look at it happening a million times, we can be almost sure to have half of them be heads,0570

because we have done it so many times that we start to see this happen more and more.0575

On a very small scale, though, we can't be certain that it will end up showing up.0579

We flip a coin three times; it might come up tails all three times, but that doesn't imply that the coin isn't fair.0583

It is just how random chance works.0588

All right, one of multiple events occurring; consider if we wanted to find the probability of rolling a fair 6-sided die and having it come up either 1 or 6.0591

Now, we could consider them as separate events;0600

so we would call them E1 and E6, the event of a 1 and the event of a 6.0601

We can talk about either E1 or E6 occurring with the notation E1 union E6.0606

That is a way of saying E1 or E6 or both.0613

What we are looking for is something that happens, which is inside of E1 or inside of E6.0616

We can be inside of either of them, so it is a union.0622

What we are curious to know here: we are looking for the probability of rolling a 1 or a 6.0625

We are looking for the probability of E1 union E6, the probability of E1 or E6 or both of them.0630

Notice that E1 and E6 are mutually exclusive events.0638

If one of them occurs, the other one cannot occur.0642

What this means is that, if we roll a 1, it is impossible to have rolled a 6 just then.0646

If we rolled a 6, it is impossible to have rolled a 1 just then.0650

There is no overlap between them; we can't be a 1 and a 6 simultaneously.0653

So, they are mutually exclusive events.0657

With this in mind, we see that the probability of E1 or E6 occurring, E1 union E6, just combines their probabilities.0659

The probability of E1 union E6 is equal to the probability of E1, plus the probability of E6.0667

Because we don't have to worry about them overlapping, it is just "did E1 happen?" and then we also could look at "did E6 happen?"0673

The probability of E1 is 1 out of 6; the probability of E6, rolling a 6, is 1 out of 6.0680

We add those two together, and we get 1/3 as the total probability for what we would have of rolling a 1 or a 6.0685

This idea works in general: given two mutually exclusive events, A and B,0694

the probability of either one, or both, occurring, A union B, is given by:0700

the probability of A union B is equal to the probability of A, plus the probability of B.0704

Given that two events can't both happen simultaneously, if we know that, if you are A, then you can't be B,0710

and if you are B, then you can't be A, then if we are looking for either one of them happening,0717

it is just going to be adding the two probabilities together.0722

This is another one of those basic ones that you end up seeing on tests and homework.0724

This is a good one to remember.0729

What if the events are not mutually exclusive--there is some overlap in the events, so that you could be in A and B at the same time?0732

For example, let's consider the probability of a fair die coming up strictly below 4 and/or (so it can also be) coming up even.0740

So, the die comes up below 4 (that is 1, 2, 3), or the die comes up even.0751

Notice: there is some overlap in these two events.0757

The die could come up as 2, which is below 4, and also which is even.0761

It is both of these at the same time; it is both of the things.0768

Since it is both of the things, there is overlap between being below 4 and being even.0772

To find the probability, we have to take this overlap into account.0777

In two events, A and B, we denote the overlap of both occurring at the same time as0781

A intersect B, that is to say, where A and B are happening at the same time.0786

So, where A intersects with B is their area of overlap, where both things are happening at the same time.0792

We take the possibility of overlap into account as follows: let A and B be two events.0800

The probability of A or B or both is given by probability of A union B, that is to say, A or B, or both occurring,0804

is equal to the probability of A, plus the probability of B, minus the probability of A intersect B.0813

That is, minus the probability of A and B occurring at the same time.0820

So, this is an interesting formula; but that one that we just talked about, where we assumed that they are mutually exclusive--0825

that one is much more likely to come up in homework and tests.0830

You might end up seeing the formula that we are working on right now.0834

But it is less likely; so if this one doesn't make quite as much sense to you, don't worry about it too much.0838

We are about to see a quick example, though, that will help explain it.0842

So, that might help cement it; but don't worry about it too much if this one doesn't make a lot of sense.0845

You are much less likely to see it than the previous one.0848

Our previous example, this example that we are looking for, 1, 2, or 3, or an even number, or both of them:0852

we can let E123, which is the event of rolling a number strictly below 4,0858

which is to say a 1, a 2, or a 3; and then Eeven, which is rolling an even number (you roll a 2, a 4, or a 6)...0863

now notice where E123 and Eeven overlap.0873

These two things overlap at E2, when we roll a 2.0878

So, if a 2 comes up on the die, it is below 4, and it is even.0884

So, E2 is where they overlap each other.0889

By the above formula, the probability of A union B, the probability of E123 union Eeven,0893

of having it be either below 4 or even or both, is equal to the probability of the die coming up as 123,0898

plus the probability of the die coming up even, minus the probability of the die coming up both of these0909

(the probability of it being 123 and even).0916

So, what is the probability of it coming up as 123?0920

Well, 1, 2, 3...that is 3 possibilities, divided by 6 total on the thing, so it is 3/6.0922

Plus coming up even: 2, 4, 6; that is 3 possibilities, divided by 6 total, so 3/6 as well.0928

Minus...and now we can swap, since we know that E2 is the same thing as E123 intersect Eeven.0935

So, it is the same thing as just asking what the probability is of rolling a 2.0940

We combine red and blue together, and they come to be a 1; 3/6 + 3/6 is 1.0944

And then minus...what is the probability of rolling a 2?0951

Well, that is 1 (rolling a 2), divided by 6; so we have 1/6; 1 - 1/6 comes out to be 5/6.0954

So, we have 5/6 as the chance of rolling a number that is below 4, or even, or both.0963

Now, 5/6 makes sense, because we can also just go through this and do this by hand.0970

1, 2, 3, 4, 5, 6: notice: if you are 1, 2, or 3, then these ones are good; if you are even: 2, 4, and 6--then those ones are good, as well.0974

So, the only one that fails to be 1, 2, 3, or even is the number 5; that means we have0993

1, 2, 3, 4...1 here; 2 here; 3 here; 4 here; 5 possibilities for this event to occur,0999

divided by a total of 6 possible outcomes; so 5 divided by 6 is the exact same thing, so this checks out and makes sense.1008

So, this formula here makes sense on the small scale, and we can also bring it to a much larger scale,1016

if we are working with a much more complicated problem, where we can't just do this by hand,1021

and we have to be able to understand the theory to be able to get an answer.1024

All right, what if an event doesn't occur?1029

If we have some event E, we can also talk about the event of E not occurring.1031

We have E that occurs when the event E occurs; but we can also talk about if E does not occur; let's make that an event.1035

E not occurring is now an event, as well; we call this the complement of E.1042

The complement of an event is that event not occurring, and we denote it with E with a little c in the top corner,1047

so E to the c, but not actually raising it like an exponent; it is not the same thing...E complement.1054

Other textbooks or teachers might denote this as E with a bar on top or E with a little tick mark.1060

It doesn't matter; any of them is fine; but I am going to use Ec.1066

For example, if E is the event of rolling an even number on a die, then Ec is the opposite event.1070

In this case, we have that E is an even number; so Ec would be the complement,1077

when E does not occur; so if E does not occur, what is the opposite to rolling an even number?1082

That is rolling an odd number; so if you roll an odd number on the die, you are in the complement of the event here.1089

The probability of an event's complement occurring is 1 - the probability of the original event.1096

So, the probability of an event's complement is equal to 1 minus the probability of the original event.1102

Why? Well, either E occurs, or it does not; and if it does not, then we know that Ec must occur.1109

So, Ec must occur if we end up having that E does not occur.1118

If E occurs, then E has occurred; if E does not occur, then Ec has occurred.1124

So, no matter what, we can be certain that one of them must occur.1130

One of these two things always has to happen.1134

We can't have something in the middle: either the event happens, or the event doesn't happen.1137

So, if the event happens, or it doesn't happen, well, either way, one of those two things happened.1141

So, we are certain that one of them will occur.1146

Since one of them always has to occur, that means that the total of their probabilities must be a 1, certainty.1150

The probability of an event's complement, plus the probability of the event, is equal to 1,1156

because one of those two things always has to happen.1160

So, that is why we end up having 1 minus the probability of the event give us the probability of the event not happening.1164

This is a useful idea in a lot of situations.1170

So, this is another useful one to remember.1172

Independent events: consider rolling a die and flipping a coin.1175

How can we find the probability of the die coming up as a 5 or a 6, and the coin coming up heads?1179

To do this, we must consider the probabilities of both events.1186

In the example of the above events, we say that they are independent events,1189

because rolling the die has no effect on flipping the coin, and flipping the coin has no effect on rolling the die.1193

They are separate events, and the outcome of one event does not affect the other.1198

We know that they are independent events in this case.1203

If they are independent events, given two independent events, A and B, the probability of both events occurring,1206

that is to say, the probability of A and B occurring, is equal to the probability of A, times the probability of B.1213

We multiply the probabilities of each of them on their own, and that gives us the probability of both of them occurring, if they are independent events.1222

In the above example, we would have: the probability of an event of a 5 or 6, and the event of a heads,1230

is equal to the probability of the event of a 5 or a 6, times the probability of the event of a heads.1239

Well, what is the chance of a 5 or a 6?1244

A 5 or a 6 is going to be 2 out of 6; and the probability of a heads is going to be 1 out of 2.1246

We multiply these together, and we get 1/6.1253

This right here, this idea of multiplying probabilities together, is the other really important idea for this lesson.1259

That very, very basic one of the number of ways that the event can happen, divided by total number of things that can occur--1266

that is the first really important idea in this lesson, and the second one is:1272

if you get independent events, and you want to figure out the chance of both of them occurring,1275

you just multiply the probabilities of each of them on their own.1279

That one also comes up on tests a lot; that is another very important idea to take away from this.1282

What if the events are not independent, though--what if we are looking at a situation1287

where they aren't actually going to not affect each other, where they can have an effect on each other?1290

The outcome of one will do something to the outcome of the other, or at least the chances of the outcome of the other.1296

For example, consider of we draw two cards at random from a deck of 52 cards.1301

What is the probability of a pair of aces--that is, both cards coming out as aces?1306

In this case, the probability of the second card being an ace is affected by what the first card was,1310

because it can change the number of aces in the deck.1317

If we pull out a card that is an ace on the first one, then there is now one fewer ace for our second pull.1319

If we don't pull out an ace on the first one, then there is the same number of aces in our second pull.1326

So, what we do on that first pull affects what will happen in the second pull.1331

We denote conditional probability with this sort of symbols, this notation.1336

The conditional probability of B occurring if A does occur is p of B bar A n things.1343

So, that is how we would say it--just the symbols.1352

If we want to talk about it, it would be the conditional probability of B occurring if A occurs.1354

Or we could also say this as the probability of B occurring, assuming that A occurs.1358

So, if we can know for sure that A will occur, then what is B's chance of occurring, with that piece of information already in mind?1363

This is the idea of conditional probability.1371

We assume the second one; the second one is the assumed one, and the first one is what we are looking at.1374

We assume the second one, p of B, A, the conditional probability of B occurring if A does, so B bar A.1391

A is going to be assumed; the second thing that shows up is assumed.1398

The first thing is now what we are trying to figure out the probability of, if we can have that assumption.1401

All right, so in the above example, we could denote the probability of the second ace being pulled, if the first ace has already been pulled.1405

What is the chance of that second ace coming out, if the first ace has already come out?1412

Then, we have our assumed thing, the first ace; and what we are looking for now is1417

the probability of that second ace, with that assumption there: the probability of the second ace, assuming a first ace.1423

Given two events A and B, where the outcome of A affects the outcome of B,1431

the probability of both events occurring is: probability of A and B is equal to the probability of A occurring on its own,1435

multiplied by the probability of B occurring, assuming that A occurs (this conditional probability).1443

Let's see this as an example: in the previous example, the result of one card1450

affects the result of the second card, because it changes what is in the deck.1453

So, the probability of that very first ace is 4 out of 52, because there are 4 aces in the deck when we pull it out, and there are 52 cards total to pull from.1457

But for the second ace, there is now one less ace, and there is one less card, if we assume that an ace was pulled on that first draw.1468

So, there is one less ace; that means that we now have 3 aces to pull from,1477

divided by 51 cards that we are pulling from total, because now there is just one less card in the deck.1481

Thus, the probability for drawing a pair of aces--if we were looking for the probability of both of these happening,1486

the first ace and a second ace pulled on two cards, then the probability of that first ace,1492

times the probability of the second ace, assuming a first ace has already been pulled, is how we get this.1498

We multiply those two together: so the probability of the first ace was 4/52, and then we multiply it1505

by the probability of the second ace, if we can assume that the first ace has already happened.1511

4/52 times 3/51...that comes out to be 1/221.1516

All right, that idea of conditional events is really interesting stuff.1524

But it is also probably the most extreme stuff that you would end up seeing on a test.1527

I would doubt that you would even see that very often, but it is pretty cool; we will see that in the final example.1532

All right, let's start with a nice, simple example: Given a bag containing 4 red marbles, 7 blue, 11 green, and 2 purple,1536

what is the chance of a blue marble being drawn randomly from the bag?1543

We are looking for what the probability is of a blue coming out.1547

How do we figure out probability? Well, we know that we are drawing randomly.1553

So, if it is randomly, we know that all of the possibilities are equally likely; that is what that "random" word means:1558

that it is a tip that says that all of these are equally likely;1563

you don't have to worry about it, which means that we can use that nice, simple formula.1566

So, it is the number of ways that the event can occur (in this case, the number of blues we could pull out),1569

the number of blue marbles, divided by the number of marbles total, all of the ways that something can happen.1575

In this case, we know...how many marbles are there that are blue? 7 are blue.1587

So, it is 7 divided by...how many marbles do we have total?1592

We have 4 red; we have to include the 7 (they are still 7, so they count as some of them), plus 7, plus 11, plus 2.1595

That is all of the marbles totaled together: we work that out: we have 7/(4 + 7 + 11 + 22), or 24; so 7/24.1605

We can't simplify it any more; and there is our probability.1615

A 7 out of 24 chance exists of pulling a blue marble randomly from the bag.1619

The second example: a class has a breakdown in grades shown by the table below.1624

If a student is selected randomly from the class (once again, there is that word "randomly";1628

it means that we can just assume that everything is equally likely), what is the chance that they have a B or a C?1632

If we are doing this, the probability of a B or a C...well, we can also look at that as the union, B union C.1638

It can be B, or it can be C; so we can add these together.1651

That would be equal to the probability of a B, plus the probability of a C, because it can be either B or C.1654

And we know that they are mutually exclusive; we don't have to worry about them being B and C simultaneously.1662

So now, we can work this out: what is the probability of a B?1667

Well, there are 12 students that have B's; let's expand a little bit more...1669

Here is the probability of B's, and then we will have the probability of C's over here.1675

How many students have B's? We have 12 students with B's, divided by...how many students do we have total?1680

We have 8 here, 12 here, 5 here, 7 here; so we add them all together to figure out how many students we have in the class.1686

8 + 12 + 5 + 7; then we add...what is the probability of a C student?1692

Well, a C student has 5...there are 5 C students total in the class.1697

And then, once again, we divide it by the number of students total, 8 + 12 + 5 + 7.1701

We work this out: 12 over...8 + 12 is 20, plus 5 is 25, plus 7 is 32, plus 5/32, equals 17/32.1707

We can't simplify that any more, so there is our probability.1721

If we draw a student randomly from the class, we have a 17/32 chance of pulling a B or a C student.1724

Next, a bag of marbles contains 12 red marbles, along with various others.1733

If you draw a marble from the bag at random, you have a 15% chance of drawing a red one.1738

What is the total number of marbles in the bag?1744

So, for this, we have 12 red marbles; we know that, if we pull out of the bag at random, we have a 15% chance of drawing a red one.1746

So, our very first thing is that we want to turn 15% as a chance...we can't really work with percentages.1752

We have seen this before; we have to turn them into decimal numbers before we can work with them in math, usually.1758

15% we change into 0.15; so it is a 0.15 probability.1763

Now, if we work this our normal way, then we know that the probability of pulling out a red1769

is equal to the number of reds in the bag, divided by the number of marbles total.1775

The total number is on the bottom.1788

What is the probability of pulling out a red?1790

Well, that is a 0.15; how many red are there? There are 12 red marbles.1792

So, it is 12 divided by the number total; the number total is just some number.1798

If we wanted, we could replace it with x, or whatever we felt like.1805

I am just going to keep writing "number total," because we know that it is just a number that we are working with.1808

We multiply both sides by number total; number total will cancel out on the denominator on the right, and appear on the left.1813

And we divide both sides by 0.15; so now, we have 12 over 0.15.1818

We use a calculator to figure out what is 12 divided by 0.15; that comes out to be 80.1824

So, we know that the total number of marbles in the bag is 80 marbles, because we were guaranteed1831

that if we pull out randomly from the bag, there is going to be a 15% chance of drawing a red one.1836

And we knew how many red marbles we started with.1842

It is a probability, with just a slight spin of algebra on it.1844

The fourth example: If you roll three fair six-sided dice ("fair" means that all of the sides are equally likely),1848

what is the chance of a 6 coming up on each die? on none of the dice? and finally, on at least one of the dice?1855

So first, let's say we are looking at all dice.1862

6 on all: if it is going to be a 6 on all of them, then it is a question of what the first one is.1866

Well, the second one is an independent event of the first one.1875

And the third one is an independent event of the first two.1878

One die's result does not affect the result of the next die, which does not affect the result of the next die.1881

They are independent events; so what is the chance of one die coming up as a 6? It is 1 out of 6.1885

What is the chance of the next die coming up as a 6? It is 1 out of 6.1893

The chance of the next die coming up as a 6 is 1 out of 6.1896

By our rule about independent events, it is multiplying them together.1899

If they are independent events, and you want to know what the probability of them all occurring is, just multiply all of the probabilities together.1903

1/6 times 1/6 times 1/6...we work that out; that comes out to 1/216; so it is a 1 out of 216 chance of a getting a 6 on all of the dice.1909

Next, what if we want to do none of the dice?1921

If it is none of the dice, what is the chance of getting no 6's on any die?1926

Well, that would be 1, 2, 3, 4, 5...those are the things that are allowed to come up.1933

So, we have 5 possibilities, divided by 6, the total number of things that could happen.1937

And that is going to be the same for the second die and the third die: 5/6 times 5/6 times 5/6.1942

We could also write this as 53, divided by 63.1949

We work that out with a calculator, and we get 125...actually, we probably don't need a calculator for that...divided by 216.1953

So, the chance of getting no 6's on any dice is 125/216.1961

Finally, at least one of the dice: this might be the one that seems hardest at first,1968

but it is actually easy, once we know this part right here, if we know none of them.1974

So, remember what we talked about before with the complement of an event.1979

If an event does not happen, then we are talking about the event complement happening.1982

If E does not happen, then Ec does happen.1988

If none of the dice happens, then there are no 6's on any of them.1992

But if none of the dice does not happen--that is to say, we don't roll a 6 on none of the dice--1996

then that means we have rolled a 6 on one of the dice, or more.2002

So, at least one is going to be the probability of the complement to the none event.2005

So, what we talked about before was the probability of none, complement: it is equal to 1 - the probability of none.2010

Another way of looking at it is just that, if you know what the probability of an event is,2023

then the probability of the opposite thing happening is 1 minus the probability of that event; that is what this none complement thing is.2027

It is the opposite of that event occurring.2033

We know 1 minus...we just figured out what is the probability of none occurring; it is 125 over 216.2036

So, we get that 91 out of 216 is the chance that at least one of the dice will come up with a 6 on it.2043

The final example: A poetry class has 17 boys and 13 girls.2054

If the teacher randomly selects 4 students from the class, what is the chance that they will all be boys?2059

This is the idea of conditional probability.2065

We have some first thing that is going to happen; but then, the second thing is going to also be affected by that first thing.2068

And then, the third thing is going to be affected by that second, and the first, thing.2076

And then, the fourth thing is going to be affected by that first, second, third thing as well.2081

So, what we do is figure out the probability of the first thing; then we multiply it by the probability of the second thing, assuming that first thing.2086

And then, we multiply that by the probability of the third thing, assuming those first two things.2091

And then, we multiply that by the probability of the fourth thing, assuming those first three things.2094

That is how conditional probability worked when we talked about it earlier.2098

So, if we start with how many students there are total in the class...if we have 17 boys and 13 girls, we assume we have 30 total.2101

So, if we have 30 total, then for the first one, we have 17 out of 30.2110

But for the next one, we pull out one of the students.2116

We have pulled out one boy; so that means we now only have 16 boys.2120

How many students do we have total? Our total of students has also gone down by 1, because we have already used one of the students.2124

We pulled out a boy; he is still one of the students in the class, so we now reduce from 30 to 29 students in the class.2130

Next, we pull out another boy; we are going to now be at 15 boys, divided by...we pull out another student...28.2136

Finally, our fourth boy: we are now at 14 boys left to pull from, and we are now at 27 students total in the classroom, after our three pulls so far.2142

If we want to figure this out, conditional probability is that we just multiply them all together: 17/30 times 16/29 times 15/28 times 14/27.2151

We work that out, and that ends up simplifying to the not-that-simple-looking 68/783, which comes out approximately to 0.087.2165

We have a little bit less than a 10% chance, a .087 chance, of managing to pull all boys if we pull 4 students.2178

It drops down pretty quickly; the first boy has a 17/30 chance, but we drop down pretty quickly by the time we are at the fourth boy.2186

It is a less-than-one-in-ten chance.2192

All right, that finishes all of our stuff about combinatorics in here, our ideas of counting, permutations, combinations, and probability.2194

I hope you have a reasonable grounding; this is all of the basics that you need for this level of math.2201

But there is a huge, huge amount of stuff to explore out there.2204

If you thought that this stuff was interesting, just do a quick search on combinatorics.2207

You will find out all sorts of cool things; there are all sorts of really cool things in combinatorics--2210

how to count things; there are lots of cool ideas in that.2214

All right, we will see you at Educator.com for the next lesson--goodbye!2216

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