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Post by Brijesh Bolar on August 17, 2012

You are so funny at times..

Conditional Probability

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  • Intro 0:00
  • Roadmap 0:05
    • Roadmap
  • 'or' vs. 'and' vs. Conditional Probability 1:07
    • 'or' vs. 'and' vs. Conditional Probability
  • 'and' vs. Conditional Probability 5:57
    • P (M or L)
    • P (M and L)
    • P (M|L)
    • P (L|M)
  • Tree Diagram 15:02
    • Tree Diagram
  • Defining Conditional Probability 22:42
    • Defining Conditional Probability
  • Common Contexts for Conditional Probability 30:56
    • Medical Testing: Positive Predictive Value
    • Medical Testing: Sensitivity
    • Statistical Tests
  • Example 1: Drug and Disease 36:41
  • Example 2: Marbles and Conditional Probability 40:04
  • Example 3: Cards and Conditional Probability 45:59
  • Example 4: Votes and Conditional Probability 50:21

Transcription: Conditional Probability

Hi and welcome to www.educator.com.0000

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 is 0103

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 know 0154

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 that 0311

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 2000 0599

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

Notice that you might think that the probability of being a male given laptop 0801

is the same as the probability of laptop given male but these are actually two different numbers. 0809

Why is that?0814

Let us think about this. 0816

Here is my little world of male and laptop, in this case my whole universe is this red universe, and I want to know, 0818

What is the probability of male given this universe given the laptop universe?0841

However, in this case I have a new universe that I'm working with.0850

I am working with the universe of given that they are a male, what is the probability that they own a laptop?0864

Also this shaded in number is the same right, this 1200 is the same as this 1200, this box right here.0872

Even that is the same the relative world that I'm looking at it in is different.0880

Here there are 1600 males, here there are 1600 people who own laptop.0887

Here are 1500 males and because of that changes are proportion.0893

They change the total and they changed proportion.0898

That is the difference between those four things that you can see that they are intimately interconnected but there is actually also a different way of looking at it.0905

You could also picture the same table as a tree diagram and this is just another way for us to be able to picture conditional probability.0916

It does not matter which one you choose first, which variable you choose whether gender or the computer ownership.0929

I will just start with gender.0938

I am going to put gender here and then put in a computer here.0944

At the end I am going to talk about what is the actual event with that combination called in.0949

It looks sort of familiar from some of the probability space stuff we have looked at before. 0954

First we need to know okay, what is the probability of being a male and what is the probability of being a female in this universe?0960

In this universe, the probability of being of male you can find down here, 1500 out of 2000 and so the probability of being a male is 75%.0969

On the flip side, the probability of being female is 25% and together this makes a total of 100%.0986

Notice that these are not yet conditional probability they are just regular old probabilities.0998

We already know how to do this but if we knew that they were a male if we already knew this person is a male then how would we figure out if they own a laptop or desktop. 1004

What is the probability that they own a desktop given that we already know that they are male?1021

Here we compute conditional probabilities.1026

Given that we know that they are male, what is the probability that they own a desktop?1029

We want to reduce our world to just the male world now.1037

What is the probability that they own a desktop?1047

That is 33%, 500 out of 1500.1050

Here we find that this is our new world we are not using the 2000 number anymore.1060

We are using the smaller space just the male space. 1067

Now, what is the probability that they own a laptop given that there are male?1071

These 2 should add up to 1 and indeed it does. 1076

It is 1000 out of 1500, so that is .6 repeating.1080

Here we have at the end of this we know that they are male and they own a desktop.1093

Here we know that they are male and they own a laptop.1101

It is like you follow this little tree, you follow that little tree.1106

Let us talk about females. 1111

Given that you know that there are female, now we are talking about this universe. 1113

Given that you know that they are female, what is the probability that they own a desktop?1118

Desktop given female, this is my new universe and it is out of 500.1125

We have 100 out of 500 students owning a desktop and that is 20%, and what is the probability that they own a laptop given that there are female?1133

Well, these 2 should add up to 100% it should be 80% and so it is 400 ÷ 500 and so that is 4/5 05 80%.1149

Here we know that once you go through this branch, this path, we know that your female and own a desktop and here we know that your female and laptop. 1161

By the end of this you get these four different events, but these four different events have very different probabilities. 1180

It is not like before, head or tails or we could just all assign them the same probability 1187

but not only that, but if we look down deeper we know that these are different than these.1190

Because these are all out of the total number of student, but this was out of whatever condition is being that.1200

How do we find the probability of male and desktop that now is not a conditional probability, I just want to know this number.1216

If you have the entire table it is actually quite easy because if you would look at male and desktop, 500 out of 2000 because it is out of the world.1230

It is not a conditional probability, so that is 25%.1238

What is the probability of male and laptop.1244

You would look at this one out of 2000 that is 50%. 1249

Notice that these two do not add up to 100%. 1254

You actually have to count all four of these in order to add up to 100%. 1258

What is the probability of female and owning a desktop, that 1 or 100 out of 2000.1264

That is going to be 1 out of 20, that is 5%.1272

What is the probability of females and laptops?1283

That is 4 out of 20, that is 20%.1287

Here what we see is when you add all of these up then you get 100%. 1305

This indeed is the entire space 25 + 50 + 25 = 100%.1315

Additionally here you add these up you get 100% because these are like subspaces of the universe, but this one is out of the whole universe.1324

Let us have breaks down.1337

Notice that these are all different, that this probability of desktop given male is not the same as the probability of male and desktop. 1338

This one is out of the entire universe.1350

This one is out of this limited universe of already knowing that there are male.1353

That is the tree diagram.1360

These are going to be the same numbers and I just rewrote them in a nice clean computerized sort of way. 1364

Here is my world where there you know that their male. 1374

Here is my world where you know that they are female. 1378

In these conditions this adds up to 1, this adds up to 1 but here is the thing, what is the relationship between these events and these guys?1382

Here when we come to events we are talking about the probability D and M and the probability of laptop and male.1394

Is there a relationship between conditional probability and “and” probability?1407

Actually there is and because if we remember what this was, the probability of desktop and male was 500 out of 2000.1414

Let us look at that relationship.1434

Here we were putting together the probability of that middle overlaps space over 2000 and that is different than this one which is .33.1437

and is also different from the one which is .75.1452

Here is what I want you to see, let us play around with this.1467

If we multiply these two together, we are taking 1/3 of 75.1472

I will multiply it like this.1483

This is what you get about 25% and it seems that this rule seems to work for every single one of these.1487

The way that I picture, let me draw for you my male world and draw in sort of a roughly proportional way.1496

Hopefully you can see that that is about like 75% this is 25%. 1521

That is my female and I know that the probability of desktop given male is 33%.1533

What I want to do is I want to divide this thing up into thirds and I want 33% of that 75%.1547

In order to get that I just multiply.1560

I just multiply 75 times the third and then I get whatever that actual area is. 1563

That area is indeed 25%. 1570

It is 1/3 of .75.1577

That actually works for all of these things. 1581

For instance here we have 2/3 of 75 and that is going to be this other section.1584

In order to get that I just take 2/3 of 75.1595

If you remember back in the day you could just think about it that is like taking 2/3 of 75, and multiplying.1603

That gives us sort of the raw value for that and that is 50%.1612

Let us see if that works with this one.1619

This is 80%, 80% of 25.1623

Here we want this much of 25%. 1629

What is the raw value of that?1635

Here what you would do is just multiply 80% × 25%, and we get 20%. 1639

This section is 20%. 1650

And then it makes sense that that is .05%.1661

Here we see that in order to find this probability all we need to do is multiply the two probabilities together right. 1663

Let us read that out.1672

The probability of D and M desktop and male is given by multiplying this proportion of the space. 1674

This is like the total space and you multiplied by this proportion.1687

The probability of D given M times the probability of M.1691

If we want to generalize this, that it is not just for males and desktop, 1699

we might just write it in terms of you know A and B equals the probability of A given B times the probability of B.1704

My use of A and B is arbitrary and you can put x and y or b and z or whatever you want.1716

It is just saying that you can generally follow the pattern and in fact you could also have the probability of B given A 1725

so the proportion of the A space that is occupied by B given the entire A space times the probability of B.1739

This is sort of the generalized form of this part, and so now we know that there is a mathematical relationship between conditional probability and the “and” probability.1748

If we wanted to get to it what we can see is that one of the very definition of conditional probability because we are defining conditional probability. 1766

Well, if we take this rule of probability of A and B equals the proportion of A knowing that B out of the B space times the whole proportion of B.1786

We can rewrite this so that instead we solve for conditional probability that would be probability of A given B equals I want to isolate this 1804

and divide both sides by probably of B.1814

Equals the probability of A and B all over the probability of B and that would give us conditional probability every time.1823

The probability that both of these are true over the probability of just B being true.1832

Here is our definition of conditional probability, mathematical definition.1838

It is very reasonable. 1849

Let us talk about some common context for conditional probability.1856

In condition probability one of the like I still hear this all-time whenever I hear like drug commercials and things like that 1862

or even you know when you go to the doctor's office and you have a test they might tell you some of this information.1871

In a medical testing, it is not necessarily that you have a disease just because the tests says that you have a disease.1878

There might be a false alarm there.1886

They might say, oh, you have the disease, but actually you do not or pregnancy test is another example where you might say that you are pregnant, 1888

but it is actually just a false alarm.1896

On the flip side, you can have a list where you actually have a disease, but the tests turns out to be negative.1899

There are all these issues with medical testing and because of that conditional probability is important.1908

When you have some sort of medical screening you want to have positive predictive value. 1914

This means that given that the test is positive, when you are the patient all you know is the test results.1922

Given that the test is positive, what is actual probability that you have a disease?1933

There is the probability of disease given that you have already got a positive result.1943

Is it 100%? Is it 90%? Is it only 10%?1948

As this positive predictive value we might also collect ppb as this goes up, that is one measure of how good your test is.1955

You want this to be high if you get a positive result is definitely you want to have the disease, but you definitely want to know if this means you have disease.1968

That alone is not enough, it is not enough just to have high ppb. 1978

You also need what is called high sensitivity and this is the opposite conditional probability.1987

This is the probability given that you have the disease and that we do not know what is the probability that the test is positive?1992

And the other less condition, what is the probability that the test is positive?2008

Imagine let us say people have this disease, the test does not only show that you have the disease.2013

That is a very different space than the probability of disease given the test results positive.2023

We need to know both of these things in order to figure out whether a medical test is any good or not. 2030

Okay, so that is one area where conditional probability turned out to be very important, 2038

and understanding that these two things do not equal each other is important. 2046

Oftentimes doctors might just give you one of these but usually it is this one.2050

Doctors or companies might tell you this one but they might not tell you this one but you need to know both in order for you to make informed decision.2057

The other place where conditional probably is very commonly used in statistical testing.2065

In statistical testing, we often have some data, and we want to know is this model likely.2074

Here we might want to note what is the probability of me getting this data during the empirical sample, sample data given some theoretical model population?2084

We had some sort of model like people are guessing randomly or this is a fairer dice or whatever our model of the universe is.2101

There is no difference between these two groups of people. 2116

Whatever our model is we want to know, what is the probability that I will get particular set of data?2125

It is good to know this but we actually also would like to know the flip side, in order to make informed decisions.2132

The probability of a particular model given a sample data and in fact we would like to know because we do already have the sample data. 2144

We want to know how likely is our model now that we are ready have our sample data, but this is actually impossible to calculate.2160

Because in order to calculate this you would have to have a model in the first place.2168

Usually we use this, but in an ideal world, we would want to know this as well,2175

but this is often an unknowable in statistics and there are some cases where they use this model thing, looked at how likely is it model given data,2176

but that is much more much beyond introductory level statistics. 2184

Let us move on to some examples.2204

In example 1, we have some sort of medical tests, this pharmaceutical rep shows these drug testing results to suggest that this drug is very accurate.2207

I guess you take a drug in order to take this test. 2216

He says that this test detects this very rare disease 90% of the time. 2220

Is there any reason to think this drug is not that great?2229

When he says this disease 90% of the time is he saying 9 out of the 10,000 people who took the test?2234

No, he is talking about 9 out of the 10 people. 2247

When he says 90% he is saying the probability of a positive test given that they had the disease is 90%.2253

This is what we call the sensitivity.2271

This test has good sensitivity.2274

When you definitely have a disease it highly likely that you will have positive results but is there any reason to think that this drug might not be that great?2277

We might want to calculate the positive predictive value.2288

This would be the probability of disease given a positive results.2293

In this case we are looking at this world the test is positive given this world and that would be 9 out of 59.2302

9 out of 59 is 15%.2315

15% of the time when you get a positive result you have the disease that means 85% of the time when you get a positive result you do not have the disease.2327

Imagine patient comes in then worried they have the disease and take the drug test, what should they think?2350

Should they think they definitely have the disease?2360

No, they have 85% chance that they do not have the disease but it is just a false positive.2365

There is a reason to think that this drug is not great. 2373

They have pretty poor positive predictive value, this is pretty low.2376

Just to put it into regular terms that a patient can understand, this means that even if you have positive tests, you do not know whether you have the disease. 2383

There is an 80% that you do not.2392

That is not a great tests, so maybe that is why is not that great. 2396

Example 2, suppose Mike draws marbles at random without replacement from a bag containing 3 red and 2 blue marbles.2401

This sounds like probability whenever you see red marbles and blue marbles. 2413

Remember, he is drawing them without replacing them. 2421

So whenever he draws a marble that marble is no longer in that universe.2424

Now it is conditional probably given that you do not have the marble what was the next world like.2430

Find the probability that the second draw is red given that the first draw was red.2436

Red-red sequence.2444

It might help if we talk about the first draw and then maybe we also put in second draw and I’m going to draw a tree diagram 2445

and I think I need a third draw might as well put that in right now. 2462

Let us talk about the first draw being red or first draw being blue. 2470

What is the probability that my first draw is red?2476

It has 3 red and 2 blue marbles so the probability of first draw being red is 3 out of 5 and so my probability the flip side of the first draw being blue is 2 out of 5.2484

We add this together, you get 1 because either this one of this have to happen.2506

By the time we get to the second draw here we have 5 marbles in the back. 2516

Here we only have 4 marbles in the bag. 2524

Here we only have 3 marble in the bag.2526

I’m just going to draw that to help me out.2529

What is the probability of drawing another red marble?2531

There is only 4 marbles left and we have already taken one of these guys out.2535

There are only 2 in the marbles bag so that would be probability of another red one given red is 2/4 or ½.2542

But the probability of getting blue is this probability of blue given that I just draw a red marble.2557

While here I still have my 2 blue marbles left because I do not draw the one out of 4.2576

That is 2/4.2586

Once again, these 2 out of 200.2588

Now we can look at what is the probability that my second draw is red given that my first draw was red?2592

Here I figured out that probability that is 1/2. 2604

Given my universe where the first draw is red, this is my probability of drawing another red one.2613

What is the probability that my second draw is red given that my first draw was blue?2623

Now we have to do this part.2629

What is the probability that my second draw is red?2631

Here, I know that I only have four marbles left but one of the ones that were taken out was a blue marble, I still have my three red marbles left in there. 2638

My probability of red, given but I just draw a blue one is 3 out of 4.2652

On the flip side, the probability of blue, given that I just draw a blue one is 1/2 because now I only have 1 blue left out of four marbles. 2664

Here I can say my probability of red given blue is ¾.2677

Finally, we come to see what it is the probability that the third draw is blue, given that my first and second draw is red?2685

Here I could just follow this branch of my tree, red and red, first and second one is red and red.2697

What is my probability I do not care about this red one, I care about this blue one. 2704

What is the probability that I draw a blue?2714

Only have three marbles left and I have not draw any blue ones, so there is 2 out of 3 marbles that are blue.2717

My probability of drawing blue given 2 previous red ones equals 2/3.2725

The probably of drawing red, I have already taken out 2 red ones that will be 1/3 and that will add up.2739

This right here I will write 2/3.2745

We have a pretty high chance of drawing blue even though we started off with pure blue because we have taken out all the red ones.2750

Example 3, if you draw 2 cards from the deck without replacement what is the probability that both cards will be hearts?2757

What is the probability if you replace the first card before drawing the second?2771

Maybe we have to draw a 3 again I will draw this without replacement world.2776

My first card that I draw it could either be a heart or not a heart and there are 4 suits in a deck hearts, diamonds, clubs, and spades.2784

The probability of drawing heart is 1 out of 4, or 13 out of 52.2803

My probability of drawing a heart is 1 out of 4.2809

My probability of not hearts is ¾ but once you have this, then what is the probability that the next one that I draw out will be a heart?2815

There are 52 cards in the deck, 13 × 4 now I only have 51 cards.2830

Here I have 52 total cards here I have 51 total cards.2841

I already drawn out a heart and so there were 13 hearts now there is only 12 hearts and there is only 51 cards.2847

The probability of drawing another heart given that I just draw a heart is now I do not have 13 hearts to choose from.2857

I only have 12 hearts to choose from, 12 out of 51 now my probability of drawing the another heart given a heart the numerator remains the same, 2868

because there were 3 sets of 13, 39 cards that were not hearts but now it is 51 cards total.2880

If you add these up that makes 1.2894

I know that my probability that both cards will be hearts is 12 out of 15.2902

Both cards heart without replacement 12 out of 51.2911

Okay, but what about with replacement?2920

What if I decide to put the card back?2923

Now let us redo this, here we have hearts versus not hearts and what is the probability that I draw a heart on the first try?2928

That is going to be the same ¼, this is also the same ¾.2954

Then I will put the card back, reshuffle the deck.2959

I replaced it.2962

Since I replaced it now what is my probability of drawing a heart?2968

It is the same as before because the card deck is 52 cards again. 2974

Once again 13 out of 52 is ¼, again at the same probability.2979

Probability of hearts given heart is still ¼.2986

Not heart given heart is still ¾. 2998

What is the probability if you replace the first card before drawing the second?3004

That is this one, ¼.3012

If I replaced it know the probabilities change. 3015

Example 4, a female Republican Presidential candidate received 16% of the vote from the Republican women, 3020

and 7% from Republican men given additional information that the Republican Party is 60% male, what is the probability that a Republican, 3031

randomly selected from this poll will vote for this candidate?3043

I assume received 25% of data in a poll, so this is not the real vote yet.3046

It might be helpful to ask if we draw a little table where you can draw a tree diagram as well but I will start up with the table 3056

because this information seems more like the and information than the conditional probability information.3069

Here let us say republican women and republican man and what is the probability that they say to this candidate versus no?3077

So 16% of Republican women said yes so the percentage that said no, is 84% and only 7% of the men, that is 93% so most of the men are saying no.3102

Given additional information that the Republican Party is 60% male, what is the probability that the Republican randomly selected from this poll would vote for the candidate?3120

Well, to me it helps me if I think about what the theory definition of conditional probability is.3132

Remember, we said the probability of let us say A given B is the probability of A and B over the probability B.3144

Here we know the probability of A and B, actually we know the probably B like with a probability of how many men are there 3164

as well as we know probability of how many females there are, 40% and we also know the and statements. 3183

We know that Republican women who voted yes is actually we know this part probability of voting yes given that they are women probably voting no given that there are women.3195

We do not know the probability of A and B.3213

If I go back to my question and I want to think eventually what I want to know is that the poll of Republican are men or women.3218

I just care if they voted yes.3227

I just want to know that probability and that is going to be the total probability.3230

If I want to get that I want to know the probability of female and yes added to the probability of male and yes.3236

That is going to give me the probability of yes.3251

Although I cannot figure out, I cannot figure out just the probability of yes I could break it down into these two component parts.3255

What I am going to do is I am going to use the multiplicative rule in order to find each of these things.3268

Let us find that out, probability of female and yes would be equal to the probability of yes given female times the probability of female.3275

We know that this is 40% and the probability of yes given female is 16%.3296

This is 60.16 × .4 and that is .064%.3305

Let us do the same thing with males.3321

Let me draw the line here.3325

Let us find the probability of male and yes that would be the probability of yes given male times the probability of male 3327

and so that would be 7% × 60 there is lots more man, but few percentage of them are voting for this female candidate.3345

And that is .042 about 4%. 3365

If I add .064 and .042 I would get 10.6%. 3372

What is the probability of a yes 10.6%.3388

There is another way to do this, you could think about this as a weighted average and in that way, you eventually come out to the same idea, 3401

so you could think about waiting this with 40% and waiting this with 60% and that is the roughly the same idea.3415

That is it for conditional probability for now.3426

I will come back to it when we talk about independent events.3433

Thanks for using www.educator.com. 3438