Statistics Conditional Probability

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We are going to be talking about conditional probability.0002

Previously we have covered a little bit about the probability of or.0007

We have A or B, and we talked about probability of A and B but now we are going to be talking about conditional probability.0011

These are slightly different animal, it is related, but it is also just different.0019

And then we are going to define conditional probability and in order to find conditional probability we are going to talk a little bit about the multiplication rule.0025

It is going to be mathematical way that conditional probability fits with the other probability spaces.0035

Then we can talk about 2 common uses of conditional probability and statistics.0045

A frequent one and one commonly misunderstood is in medical testing which is that it commonly misunderstood too.0050

We are also going to talk a little bit just broadly about how conditional probability is important in statistical inference.0059

Okay, first, let us talk about the difference between or and conditional probability and also hopefully you will see the similarities as well.0069

Previously we used to draw these in terms of circles but now I am going to draw it in rectangles to save some space here.0079

The probability of A or B, A and B being disjoint events that might look like this where we have A here is that probability space for A.0089

Here is the probability space for B.0100

Notice that there is no overlap and when we talk about probability of A or B we are talking about is0103

what is the probability that you will land in at least one of these shade of spaces.0113

Same thing, there is similar idea with probability of A or B but non disjoint events.0120

The only difference here is that now there are places where A and B slightly overlap.0127

Once again, when you ask about the probability of A or B you are asking about what is the probability that you will land somewhere in not shaded space.0139

When we talk about the probability of A and B and now we only want to know0154

what probability of landing in that shaded space, just the part that overlaps where both A is true and B is true.0166

Remember or is when at least one is true this is when both have to be true.0177

This is all that we know so far, but now let us talk about the probability of B given A.0186

When there is this line vertical line here you want to read that as given A.0195

A you already know it is true.0201

Here is the difference now, now you are thinking about this red space of A.0204

This is the entire world now and now we do not care about this part of the space, this extra part.0213

Now A is our entire space.0221

What is the probability that given A we have B being true?0228

There might be a portion of B out here as well, but we just do not care about it anymore because we only want to know the probability of B given A.0237

That is the entire space and interested in, and this is the total space.0251

Let us try out probability of A given B.0263

Now your B is your entire space that is the entire space that I'm interested in.0269

There might be some A out here but we just do not care about it.0276

We do not care about any of this external space here.0287

Our whole world is now B and all we care about is the probability of A given B.0292

You might want to think about within the world of B, within the given world, what is the probability of A?0300

That is conditional probability and the reason why it is called conditional probability is that0311

first you have to meet a particular condition you are shrinking down the space first.0318

Here for now only a condition where A is already true regarding our A, only in that condition do we want to know when B is true.0323

Same thing here, this is the condition now under the condition that B is true.0334

What is the probability that A is true.0340

That is why it is called conditional probability.0343

Here, these are different because now we are not dealing with this entire space out here.0346

We are only dealing with this world where some condition has already been met.0350

Let us get down into the nitty-gritty, a little bit.0357

Let us talk about and’s versus conditional probability then let us talk about it in terms of actual numbers.0363

Let us say I give you a situation where 2000 computer science majors out of college were pulled and males and females were pulled.0369

They are asked what kind of computer do you have?0379

Do you have a desktop or laptop?0382

It turns out that there are generally more males than females and their generally more people that own laptops than own desktops0385

but it also breaks down into little subcategories.0394

We will talk about or first, what is the probability that these students fall in the category of male or owns a laptop?0400

Here we think that picture you want to think of okay here is my males, there is always male and here are my laptop people.0418

All the people who own laptops but some of these laptop think people are female, and some of these males owns desktops.0433

That is okay, what we want to know is what is the probability that you will land in any one of these spaces over the entire student body that was sampled.0441

That is the entire student body that is sampled.0456

You are looking for the probability of male or laptop over all students but you also want to take out one of those overlaps because you will count this twice.0459

The probability that they are male is you want to look in the male column.0479

You do not just want to look here.0489

You want to look at the total number of males that is 1500/2000.0491

Because what I am looking for is the probability of male plus because I am using my generalize addition rule,0502

and because these are non-disjoint events we also need to subtract the probability of male and laptop.0511

Since we need that let us start over here because we need probability of male and laptop.0523

Here what is the probability that they are both male and they own a laptop.0528

We want to look in the male and owns a laptop that is 1200 out of the entire student body.0535

It is just this overlap out of the entire box, entire space.0545

That is 60% of students are both male and own laptops.0554

Now that we know that it might be easier to do this then we want to add in percentage of people who own laptops in general.0577

Here we want to look across our costs rows and we see here there are 1600 that own laptop.0588

Notice that here, if you add these together, you are going to have a number that is over 20000599

and you cannot possibly have this little colored space be more than the entire box.0607

What we need to do is take out the overlap and the overlaps are these 1200 people.0616

If we look at this what I am going to do is take 1200 out of one of these guys so that is 400 and that here so that is 1900/2000.0628

If I want to turn that into a percentage that is roughly 19/20 so that is 9.5/10.0642

That is 95%.0652

Here we see that 95% of the student are either male or have laptops is a lot.0657

Here we see the difference between the or statement as well as the and statements.0664

Now that we know that and those are just things we already know so far.0673

Now that we know that let us start talking about conditional probability.0679

Conditional probability we want to know okay people who already meet this requirement of having a laptop under those conditions only.0684

Now our world is limited to this.0693

Now pretend we put x’s over everything else, we only care about this world, people who own laptops.0699

What is the probability of being a male given that they own a laptop?0708

Instead of using this 2000 I’m going to use my 1600 because that is my entire space now and that probability is 1200/1600.0714

That is 6/8 or 3/4 or 75%.0728

The probability of male given laptop is 75%.0740

Let us look at different condition we are now looking at the probability that they have a laptop given the condition that they are already males.0747

We already know we are limiting our world to here and pretend we put x over everything else.0761

We want to know given this little universe what is the probability that they own laptop?0767

That would be this number here that they are male and own a laptop over this new world.0774

It is not 2000 anymore, it is just the people who are already males and its 12/15, which is 4/5, and that is 80%.0784