Statistics Independent Events

Hi and welcome to www.educator.com.0000

We are going to be talking about independent events today.0001

We just covered conditional probability and independent events have a lot to do with conditional probability.0004

We are going to look at how they relate to each other.0013

We are going to actually define what is an independent event is mathematically.0015

Then we are going to modify the multiplication rule for a conditional probability for independent events.0019

First thing is first.0028

Independent events and how that fits together with conditional probability.0031

You can think about non independent events, non independent events means that if knowledge of one of the events of the out coming events affects0037

the probability that the other events occurs.0047

If I know that they are male does it affect my estimate of whether they own a lot or not.0051

If I know that this person is obese does it affect my estimate if they have heart disease?0058

Those are what we come down independent events, knowing one thing it changes your estimate of the second event.0065

Here what you could think about is this, conditional probabilities for knowing that given this person is obese that will change your estimate of heart disease versus0073

if you know that this person is not obese these 2 estimates of heart disease will be different.0092

They will be not equal to each other.0104

That is what we mean by the conditional probabilities are different because if you know one if the conditions that will change your estimate for the other event.0111

What are independent events?0122

This means that knowledge of one event does not change or affect the probability of the other event occurs.0124

Here the conditions of probabilities are the same.0133

The probability of heart disease given the kind of car you drive equals red.0137

If you drive a red car versus the probability of heart disease, this probability should be the same because car color has nothing to do with heart disease.0149

You might say this is independent, car color does not have any varying in my estimate of probability of you having a heart disease.0172

Let us talk about this a little bit more mathematically.0191

Let us use this example of obesity and gender.0196

Here is obesity on this side and gender on this side.0201

Does the probability of being male or female change whether you might be obese?0206

Does knowing whether somebody is male or female, does knowing their gender affect the probability that they might be obese?0221

It turns out that the probability of being obese given that is 20 out of 100, that is the condition of probability.0229

We are only looking at the square root.0241

The probability of obese given in a female is also 20 out of 100.0247

Here you could see that these probabilities equal each other.0257

The probability of being obese for male or female are the same.0263

How about education?0272

Is the probability of being obese given that they have a post high school education is that going to be different than the probability of being obese given0275

that they only have a high school education.0289

The probability of obesity given post high school is 20 over 100.0293

This is my universe is 20/100.0304

That probability of being obese given high school only is 30 out of 100.0307

It is higher for people who have not have post high school education than people who have had post high school education.0319

These conditional probabilities do not mean that this causes this, it is just knowing one fact about these person helps you estimate their obesity probability differently.0329

Here you can see these are not equal to each other.0346

Let us define what are independent event is.0354

We have already talked about this.0358

One way to define it is that the probability of A given B is equal to the probability of A given not B.0361

Those 2 equal each other.0372

It does not matter whether B occurs or not.0374

That is what I have written down here.0377

There is another way that you could think about this.0381

The probability of A given B is equal to just the probability of A in any circumstances.0384

This is another way of defining independent events.0392

Let us look at that with this data set.0399

What is the probability of obesity given male and is that the same with the probability of general obesity?0401

Let us calculate that.0414

The probability of obesity given male is 20 out of 100 but the probability of obesity over r is here.0415

It is obesity of all the people in this sample.0428

It is 40 out of 200.0432

That is exactly the same proportion 20%.0435

Here we see this.0439

Being male or obese is independent events in this example.0443

Now that we know how to define independent events mathematically let us talk about the multiplication rule for conditional probability.0451

Remember those trees that we found was that if you wanted the probability of A and B that is equal to the proportion of A given B multiplied by the proportion of the probability of B.0462

Think about these spaces.0484

B and what proportion of that is A given B if you multiply this together you will get that raw score.0486

This is what we call the multiplication rule for conditional probability.0494

Out of this you could also get the definition of conditional probability where probability of A given B equals the probability of A and B over the probability of B.0498

We already know the multiplication rule and that is just one step around.0516

I should have to say here that obviously you could have probability of B given A × p(A) because you always want to have that entire world that you are living in.0520

The condition that you are living in.0536

Independent events we now have a slight change than this because the probability of A given B equals the probability of A, look at this rule again.0538

All we have to do now is this, in order to find the probability of A and B since this equal this we can now just do probability of A × p(B).0550

This is exactly equal to the p(A) given B.0570

For independent events we can simplify this.0575

This all goes back to the multiplication rule.0578

In independent events, now you could just write p(A) and still be able to calculate p(A) and B.0582

The other way that you could think about this is you could change it into figuring out different relationships among these things but you can also generalize it to more than just 2 events.0593

We could put 3 events, p(ABC) = p(A × B × C).0611

You could do 4 events, 5 independent events because you can do this infinite times.0624

That way I like to think about this is going back to the sample spaces thinking about this independent events as slots that you could fill.0633

Let us think about flipping a coin, those are independent events.0643

Knowing that you first flip is a head does not do anything for my next flip of coin.0646

It is still a 50-50 chance of getting heads.0654

Here you could think about this as the probability of A, probability of getting heads is 50%, 50%, and 50%.0657

You could see that it will go on and on and on.0665

Flipping coins are classic examples of independent events.0668

Let us move on to some examples.0673

Here is example 1, suppose you draw a card at random from a deck of cards which of these pairs of events are independent?0675

You are just drawing 1 card and just because they say events it does not mean you are drawing 2 cards.0682

It just means that it is 2 different aspects of cards like heart and jack.0689

Here it says it is getting a heart independent of getting a jack.0695

Does having any of the one affect the probability of getting the other?0701

We could line out the rule for independence of events.0704

Probability of heart given jack should equal the probability of hearts.0712

Is that true? Let us think about this.0720

There are only 4 jacks, that is my whole universe and the probability of getting a heart is ¼.0724

That is the probability of getting a heart overall.0730

I would say these are independent.0734