WEBVTT mathematics/statistics/son
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Hi and welcome to www.educator.com.
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We are going to be talking about conditional probability.
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Previously we have covered a little bit about the probability of or.
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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.
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These are slightly different animal, it is related, but it is also just different.
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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.
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It is going to be mathematical way that conditional probability fits with the other probability spaces.
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Then we can talk about 2 common uses of conditional probability and statistics.
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A frequent one and one commonly misunderstood is in medical testing which is that it commonly misunderstood too.
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We are also going to talk a little bit just broadly about how conditional probability is important in statistical inference.
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Okay, first, let us talk about the difference between or and conditional probability and also hopefully you will see the similarities as well.
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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.
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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.
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Here is the probability space for B.
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Notice that there is no overlap and when we talk about probability of A or B we are talking about is
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what is the probability that you will land in at least one of these shade of spaces.
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Same thing, there is similar idea with probability of A or B but non disjoint events.
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The only difference here is that now there are places where A and B slightly overlap.
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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.
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When we talk about the probability of A and B and now we only want to know
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what probability of landing in that shaded space, just the part that overlaps where both A is true and B is true.
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Remember or is when at least one is true this is when both have to be true.
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This is all that we know so far, but now let us talk about the probability of B given A.
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When there is this line vertical line here you want to read that as given A.
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A you already know it is true.
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Here is the difference now, now you are thinking about this red space of A.
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This is the entire world now and now we do not care about this part of the space, this extra part.
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Now A is our entire space.
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What is the probability that given A we have B being true?
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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.
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That is the entire space and interested in, and this is the total space.
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Let us try out probability of A given B.
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Now your B is your entire space that is the entire space that I'm interested in.
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There might be some A out here but we just do not care about it.
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We do not care about any of this external space here.
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Our whole world is now B and all we care about is the probability of A given B.
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You might want to think about within the world of B, within the given world, what is the probability of A?
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That is conditional probability and the reason why it is called conditional probability is that
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first you have to meet a particular condition you are shrinking down the space first.
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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.
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Same thing here, this is the condition now under the condition that B is true.
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What is the probability that A is true.
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That is why it is called conditional probability.
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Here, these are different because now we are not dealing with this entire space out here.
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We are only dealing with this world where some condition has already been met.
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Let us get down into the nitty-gritty, a little bit.
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Let us talk about and’s versus conditional probability then let us talk about it in terms of actual numbers.
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Let us say I give you a situation where 2000 computer science majors out of college were pulled and males and females were pulled.
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They are asked what kind of computer do you have?
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Do you have a desktop or laptop?
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It turns out that there are generally more males than females and their generally more people that own laptops than own desktops
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but it also breaks down into little subcategories.
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We will talk about or first, what is the probability that these students fall in the category of male or owns a laptop?
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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.
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All the people who own laptops but some of these laptop think people are female, and some of these males owns desktops.
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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.
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That is the entire student body that is sampled.
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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.
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The probability that they are male is you want to look in the male column.
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You do not just want to look here.
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You want to look at the total number of males that is 1500/2000.
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Because what I am looking for is the probability of male plus because I am using my generalize addition rule,
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and because these are non-disjoint events we also need to subtract the probability of male and laptop.
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Since we need that let us start over here because we need probability of male and laptop.
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Here what is the probability that they are both male and they own a laptop.
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We want to look in the male and owns a laptop that is 1200 out of the entire student body.
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It is just this overlap out of the entire box, entire space.
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That is 60% of students are both male and own laptops.
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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.
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Here we want to look across our costs rows and we see here there are 1600 that own laptop.
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Notice that here, if you add these together, you are going to have a number that is over 2000
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and you cannot possibly have this little colored space be more than the entire box.
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What we need to do is take out the overlap and the overlaps are these 1200 people.
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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.
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If I want to turn that into a percentage that is roughly 19/20 so that is 9.5/10.
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That is 95%.
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Here we see that 95% of the student are either male or have laptops is a lot.
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Here we see the difference between the or statement as well as the and statements.
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Now that we know that and those are just things we already know so far.
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Now that we know that let us start talking about conditional probability.
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Conditional probability we want to know okay people who already meet this requirement of having a laptop under those conditions only.
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Now our world is limited to this.
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Now pretend we put x’s over everything else, we only care about this world, people who own laptops.
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What is the probability of being a male given that they own a laptop?
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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.
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That is 6/8 or 3/4 or 75%.
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The probability of male given laptop is 75%.
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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.
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We already know we are limiting our world to here and pretend we put x over everything else.
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We want to know given this little universe what is the probability that they own laptop?
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That would be this number here that they are male and own a laptop over this new world.
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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%.
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Notice that you might think that the probability of being a male given laptop
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is the same as the probability of laptop given male but these are actually two different numbers.
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Why is that?
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Let us think about this.
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Here is my little world of male and laptop, in this case my whole universe is this red universe, and I want to know,
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What is the probability of male given this universe given the laptop universe?
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However, in this case I have a new universe that I'm working with.
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I am working with the universe of given that they are a male, what is the probability that they own a laptop?
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Also this shaded in number is the same right, this 1200 is the same as this 1200, this box right here.
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Even that is the same the relative world that I'm looking at it in is different.
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Here there are 1600 males, here there are 1600 people who own laptop.
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Here are 1500 males and because of that changes are proportion.
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They change the total and they changed proportion.
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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.
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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.
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It does not matter which one you choose first, which variable you choose whether gender or the computer ownership.
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I will just start with gender.
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I am going to put gender here and then put in a computer here.
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At the end I am going to talk about what is the actual event with that combination called in.
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It looks sort of familiar from some of the probability space stuff we have looked at before.
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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?
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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%.
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On the flip side, the probability of being female is 25% and together this makes a total of 100%.
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Notice that these are not yet conditional probability they are just regular old probabilities.
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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.
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What is the probability that they own a desktop given that we already know that they are male?
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Here we compute conditional probabilities.
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Given that we know that they are male, what is the probability that they own a desktop?
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We want to reduce our world to just the male world now.
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What is the probability that they own a desktop?
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That is 33%, 500 out of 1500.
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Here we find that this is our new world we are not using the 2000 number anymore.
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We are using the smaller space just the male space.
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Now, what is the probability that they own a laptop given that there are male?
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These 2 should add up to 1 and indeed it does.
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It is 1000 out of 1500, so that is .6 repeating.
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Here we have at the end of this we know that they are male and they own a desktop.
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Here we know that they are male and they own a laptop.
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It is like you follow this little tree, you follow that little tree.
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Let us talk about females.
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Given that you know that there are female, now we are talking about this universe.
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Given that you know that they are female, what is the probability that they own a desktop?
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Desktop given female, this is my new universe and it is out of 500.
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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?
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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%.
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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.
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By the end of this you get these four different events, but these four different events have very different probabilities.
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It is not like before, head or tails or we could just all assign them the same probability
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but not only that, but if we look down deeper we know that these are different than these.
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Because these are all out of the total number of student, but this was out of whatever condition is being that.
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How do we find the probability of male and desktop that now is not a conditional probability, I just want to know this number.
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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.
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It is not a conditional probability, so that is 25%.
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What is the probability of male and laptop.
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You would look at this one out of 2000 that is 50%.
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Notice that these two do not add up to 100%.
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You actually have to count all four of these in order to add up to 100%.
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What is the probability of female and owning a desktop, that 1 or 100 out of 2000.
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That is going to be 1 out of 20, that is 5%.
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What is the probability of females and laptops?
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That is 4 out of 20, that is 20%.
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Here what we see is when you add all of these up then you get 100%.
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This indeed is the entire space 25 + 50 + 25 = 100%.
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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.
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Let us have breaks down.
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Notice that these are all different, that this probability of desktop given male is not the same as the probability of male and desktop.
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This one is out of the entire universe.
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This one is out of this limited universe of already knowing that there are male.
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That is the tree diagram.
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These are going to be the same numbers and I just rewrote them in a nice clean computerized sort of way.
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Here is my world where there you know that their male.
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Here is my world where you know that they are female.
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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?
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Here when we come to events we are talking about the probability D and M and the probability of laptop and male.
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Is there a relationship between conditional probability and “and” probability?
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Actually there is and because if we remember what this was, the probability of desktop and male was 500 out of 2000.
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Let us look at that relationship.
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Here we were putting together the probability of that middle overlaps space over 2000 and that is different than this one which is .33.
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and is also different from the one which is .75.
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Here is what I want you to see, let us play around with this.
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If we multiply these two together, we are taking 1/3 of 75.
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I will multiply it like this.
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This is what you get about 25% and it seems that this rule seems to work for every single one of these.
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The way that I picture, let me draw for you my male world and draw in sort of a roughly proportional way.
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Hopefully you can see that that is about like 75% this is 25%.
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That is my female and I know that the probability of desktop given male is 33%.
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What I want to do is I want to divide this thing up into thirds and I want 33% of that 75%.
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In order to get that I just multiply.
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I just multiply 75 times the third and then I get whatever that actual area is.
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That area is indeed 25%.
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It is 1/3 of .75.
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That actually works for all of these things.
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For instance here we have 2/3 of 75 and that is going to be this other section.
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In order to get that I just take 2/3 of 75.
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If you remember back in the day you could just think about it that is like taking 2/3 of 75, and multiplying.
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That gives us sort of the raw value for that and that is 50%.
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Let us see if that works with this one.
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This is 80%, 80% of 25.
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Here we want this much of 25%.
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What is the raw value of that?
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Here what you would do is just multiply 80% × 25%, and we get 20%.
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This section is 20%.
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And then it makes sense that that is .05%.
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Here we see that in order to find this probability all we need to do is multiply the two probabilities together right.
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Let us read that out.
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The probability of D and M desktop and male is given by multiplying this proportion of the space.
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This is like the total space and you multiplied by this proportion.
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The probability of D given M times the probability of M.
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If we want to generalize this, that it is not just for males and desktop,
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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.
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My use of A and B is arbitrary and you can put x and y or b and z or whatever you want.
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It is just saying that you can generally follow the pattern and in fact you could also have the probability of B given A
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so the proportion of the A space that is occupied by B given the entire A space times the probability of B.
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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.
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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.
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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.
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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
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and divide both sides by probably of B.
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Equals the probability of A and B all over the probability of B and that would give us conditional probability every time.
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The probability that both of these are true over the probability of just B being true.
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Here is our definition of conditional probability, mathematical definition.
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It is very reasonable.
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Let us talk about some common context for conditional probability.
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In condition probability one of the like I still hear this all-time whenever I hear like drug commercials and things like that
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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.
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In a medical testing, it is not necessarily that you have a disease just because the tests says that you have a disease.
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There might be a false alarm there.
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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,
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but it is actually just a false alarm.
00:31:39.200 --> 00:31:48.400
On the flip side, you can have a list where you actually have a disease, but the tests turns out to be negative.
00:31:48.400 --> 00:31:54.200
There are all these issues with medical testing and because of that conditional probability is important.
00:31:54.200 --> 00:32:02.300
When you have some sort of medical screening you want to have positive predictive value.
00:32:02.300 --> 00:32:12.900
This means that given that the test is positive, when you are the patient all you know is the test results.
00:32:12.900 --> 00:32:22.800
Given that the test is positive, what is actual probability that you have a disease?
00:32:22.800 --> 00:32:28.400
There is the probability of disease given that you have already got a positive result.
00:32:28.400 --> 00:32:35.100
Is it 100%? Is it 90%? Is it only 10%?
00:32:35.100 --> 00:32:47.900
As this positive predictive value we might also collect ppb as this goes up, that is one measure of how good your test is.
00:32:47.900 --> 00:32:58.000
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.
00:32:58.000 --> 00:33:06.800
That alone is not enough, it is not enough just to have high ppb.
00:33:06.800 --> 00:33:12.400
You also need what is called high sensitivity and this is the opposite conditional probability.
00:33:12.400 --> 00:33:28.000
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?
00:33:28.000 --> 00:33:33.600
And the other less condition, what is the probability that the test is positive?
00:33:33.600 --> 00:33:43.600
Imagine let us say people have this disease, the test does not only show that you have the disease.
00:33:43.600 --> 00:33:50.100
That is a very different space than the probability of disease given the test results positive.
00:33:50.100 --> 00:33:57.700
We need to know both of these things in order to figure out whether a medical test is any good or not.
00:33:57.700 --> 00:34:05.800
Okay, so that is one area where conditional probability turned out to be very important,
00:34:05.800 --> 00:34:10.000
and understanding that these two things do not equal each other is important.
00:34:10.000 --> 00:34:17.300
Oftentimes doctors might just give you one of these but usually it is this one.
00:34:17.300 --> 00:34:25.200
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.
00:34:25.200 --> 00:34:34.600
The other place where conditional probably is very commonly used in statistical testing.
00:34:34.600 --> 00:34:44.000
In statistical testing, we often have some data, and we want to know is this model likely.
00:34:44.000 --> 00:35:01.400
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?
00:35:01.400 --> 00:35:16.300
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.
00:35:16.300 --> 00:35:25.000
There is no difference between these two groups of people.
00:35:25.000 --> 00:35:32.400
Whatever our model is we want to know, what is the probability that I will get particular set of data?
00:35:32.400 --> 00:35:44.000
It is good to know this but we actually also would like to know the flip side, in order to make informed decisions.
00:35:44.000 --> 00:35:59.700
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.
00:35:59.700 --> 00:36:08.400
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.
00:36:08.400 --> 00:36:14.800
Because in order to calculate this you would have to have a model in the first place.
00:36:14.800 --> 00:36:16.000
Usually we use this, but in an ideal world, we would want to know this as well,
00:36:16.000 --> 00:36:24.500
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,
00:36:24.500 --> 00:36:44.500
but that is much more much beyond introductory level statistics.
00:36:44.500 --> 00:36:46.700
Let us move on to some examples.
00:36:46.700 --> 00:36:56.200
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.
00:36:56.200 --> 00:37:00.400
I guess you take a drug in order to take this test.
00:37:00.400 --> 00:37:09.300
He says that this test detects this very rare disease 90% of the time.
00:37:09.300 --> 00:37:14.200
Is there any reason to think this drug is not that great?
00:37:14.200 --> 00:37:26.700
When he says this disease 90% of the time is he saying 9 out of the 10,000 people who took the test?
00:37:26.700 --> 00:37:33.200
No, he is talking about 9 out of the 10 people.
00:37:33.200 --> 00:37:50.800
When he says 90% he is saying the probability of a positive test given that they had the disease is 90%.
00:37:50.800 --> 00:37:53.700
This is what we call the sensitivity.
00:37:53.700 --> 00:37:57.000
This test has good sensitivity.
00:37:57.000 --> 00:38:08.400
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?
00:38:08.400 --> 00:38:13.500
We might want to calculate the positive predictive value.
00:38:13.500 --> 00:38:22.200
This would be the probability of disease given a positive results.
00:38:22.200 --> 00:38:35.000
In this case we are looking at this world the test is positive given this world and that would be 9 out of 59.
00:38:35.000 --> 00:38:47.100
9 out of 59 is 15%.
00:38:47.100 --> 00:39:10.000
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.
00:39:10.000 --> 00:39:19.700
Imagine patient comes in then worried they have the disease and take the drug test, what should they think?
00:39:19.700 --> 00:39:25.000
Should they think they definitely have the disease?
00:39:25.000 --> 00:39:32.800
No, they have 85% chance that they do not have the disease but it is just a false positive.
00:39:32.800 --> 00:39:36.100
There is a reason to think that this drug is not great.
00:39:36.100 --> 00:39:43.000
They have pretty poor positive predictive value, this is pretty low.
00:39:43.000 --> 00:39:52.600
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.
00:39:52.600 --> 00:39:56.600
There is an 80% that you do not.
00:39:56.600 --> 00:40:01.000
That is not a great tests, so maybe that is why is not that great.
00:40:01.000 --> 00:40:13.400
Example 2, suppose Mike draws marbles at random without replacement from a bag containing 3 red and 2 blue marbles.
00:40:13.400 --> 00:40:20.800
This sounds like probability whenever you see red marbles and blue marbles.
00:40:20.800 --> 00:40:23.800
Remember, he is drawing them without replacing them.
00:40:23.800 --> 00:40:30.200
So whenever he draws a marble that marble is no longer in that universe.
00:40:30.200 --> 00:40:36.500
Now it is conditional probably given that you do not have the marble what was the next world like.
00:40:36.500 --> 00:40:43.700
Find the probability that the second draw is red given that the first draw was red.
00:40:43.700 --> 00:40:45.600
Red-red sequence.
00:40:45.600 --> 00:41:02.600
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
00:41:02.600 --> 00:41:10.500
and I think I need a third draw might as well put that in right now.
00:41:10.500 --> 00:41:16.500
Let us talk about the first draw being red or first draw being blue.
00:41:16.500 --> 00:41:24.200
What is the probability that my first draw is red?
00:41:24.200 --> 00:41:46.100
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.
00:41:46.100 --> 00:41:56.000
We add this together, you get 1 because either this one of this have to happen.
00:41:56.000 --> 00:42:03.900
By the time we get to the second draw here we have 5 marbles in the back.
00:42:03.900 --> 00:42:06.300
Here we only have 4 marbles in the bag.
00:42:06.300 --> 00:42:09.600
Here we only have 3 marble in the bag.
00:42:09.600 --> 00:42:11.000
I’m just going to draw that to help me out.
00:42:11.000 --> 00:42:15.400
What is the probability of drawing another red marble?
00:42:15.400 --> 00:42:22.100
There is only 4 marbles left and we have already taken one of these guys out.
00:42:22.100 --> 00:42:37.300
There are only 2 in the marbles bag so that would be probability of another red one given red is 2/4 or ½.
00:42:37.300 --> 00:42:56.200
But the probability of getting blue is this probability of blue given that I just draw a red marble.
00:42:56.200 --> 00:43:05.800
While here I still have my 2 blue marbles left because I do not draw the one out of 4.
00:43:05.800 --> 00:43:08.100
That is 2/4.
00:43:08.100 --> 00:43:12.500
Once again, these 2 out of 200.
00:43:12.500 --> 00:43:24.200
Now we can look at what is the probability that my second draw is red given that my first draw was red?
00:43:24.200 --> 00:43:33.600
Here I figured out that probability that is 1/2.
00:43:33.600 --> 00:43:42.800
Given my universe where the first draw is red, this is my probability of drawing another red one.
00:43:42.800 --> 00:43:49.200
What is the probability that my second draw is red given that my first draw was blue?
00:43:49.200 --> 00:43:51.000
Now we have to do this part.
00:43:51.000 --> 00:43:58.400
What is the probability that my second draw is red?
00:43:58.400 --> 00:44:12.000
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.
00:44:12.000 --> 00:44:23.700
My probability of red, given but I just draw a blue one is 3 out of 4.
00:44:23.700 --> 00:44:37.500
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.
00:44:37.500 --> 00:44:45.500
Here I can say my probability of red given blue is ¾.
00:44:45.500 --> 00:44:56.800
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?
00:44:56.800 --> 00:45:03.900
Here I could just follow this branch of my tree, red and red, first and second one is red and red.
00:45:03.900 --> 00:45:14.200
What is my probability I do not care about this red one, I care about this blue one.
00:45:14.200 --> 00:45:16.700
What is the probability that I draw a blue?
00:45:16.700 --> 00:45:25.600
Only have three marbles left and I have not draw any blue ones, so there is 2 out of 3 marbles that are blue.
00:45:25.600 --> 00:45:38.800
My probability of drawing blue given 2 previous red ones equals 2/3.
00:45:38.800 --> 00:45:45.100
The probably of drawing red, I have already taken out 2 red ones that will be 1/3 and that will add up.
00:45:45.100 --> 00:45:50.500
This right here I will write 2/3.
00:45:50.500 --> 00:45:57.200
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.
00:45:57.200 --> 00:46:10.700
Example 3, if you draw 2 cards from the deck without replacement what is the probability that both cards will be hearts?
00:46:10.700 --> 00:46:15.800
What is the probability if you replace the first card before drawing the second?
00:46:15.800 --> 00:46:24.300
Maybe we have to draw a 3 again I will draw this without replacement world.
00:46:24.300 --> 00:46:43.500
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.
00:46:43.500 --> 00:46:48.900
The probability of drawing heart is 1 out of 4, or 13 out of 52.
00:46:48.900 --> 00:46:55.500
My probability of drawing a heart is 1 out of 4.
00:46:55.500 --> 00:47:09.900
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?
00:47:09.900 --> 00:47:20.700
There are 52 cards in the deck, 13 × 4 now I only have 51 cards.
00:47:20.700 --> 00:47:26.700
Here I have 52 total cards here I have 51 total cards.
00:47:26.700 --> 00:47:36.800
I already drawn out a heart and so there were 13 hearts now there is only 12 hearts and there is only 51 cards.
00:47:36.800 --> 00:47:48.000
The probability of drawing another heart given that I just draw a heart is now I do not have 13 hearts to choose from.
00:47:48.000 --> 00:48:00.200
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,
00:48:00.200 --> 00:48:14.100
because there were 3 sets of 13, 39 cards that were not hearts but now it is 51 cards total.
00:48:14.100 --> 00:48:21.700
If you add these up that makes 1.
00:48:21.700 --> 00:48:31.200
I know that my probability that both cards will be hearts is 12 out of 15.
00:48:31.200 --> 00:48:40.400
Both cards heart without replacement 12 out of 51.
00:48:40.400 --> 00:48:43.500
Okay, but what about with replacement?
00:48:43.500 --> 00:48:47.700
What if I decide to put the card back?
00:48:47.700 --> 00:49:14.100
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?
00:49:14.100 --> 00:49:18.700
That is going to be the same ¼, this is also the same ¾.
00:49:18.700 --> 00:49:22.500
Then I will put the card back, reshuffle the deck.
00:49:22.500 --> 00:49:28.000
I replaced it.
00:49:28.000 --> 00:49:33.900
Since I replaced it now what is my probability of drawing a heart?
00:49:33.900 --> 00:49:39.000
It is the same as before because the card deck is 52 cards again.
00:49:39.000 --> 00:49:46.600
Once again 13 out of 52 is ¼, again at the same probability.
00:49:46.600 --> 00:49:58.300
Probability of hearts given heart is still ¼.
00:49:58.300 --> 00:50:03.800
Not heart given heart is still ¾.
00:50:03.800 --> 00:50:12.100
What is the probability if you replace the first card before drawing the second?
00:50:12.100 --> 00:50:15.200
That is this one, ¼.
00:50:15.200 --> 00:50:20.100
If I replaced it know the probabilities change.
00:50:20.100 --> 00:50:30.900
Example 4, a female Republican Presidential candidate received 16% of the vote from the Republican women,
00:50:30.900 --> 00:50:43.000
and 7% from Republican men given additional information that the Republican Party is 60% male, what is the probability that a Republican,
00:50:43.000 --> 00:50:46.500
randomly selected from this poll will vote for this candidate?
00:50:46.500 --> 00:50:56.600
I assume received 25% of data in a poll, so this is not the real vote yet.
00:50:56.600 --> 00:51:08.800
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
00:51:08.800 --> 00:51:17.300
because this information seems more like the and information than the conditional probability information.
00:51:17.300 --> 00:51:41.900
Here let us say republican women and republican man and what is the probability that they say to this candidate versus no?
00:51:41.900 --> 00:52:00.400
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.
00:52:00.400 --> 00:52:12.000
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?
00:52:12.000 --> 00:52:24.600
Well, to me it helps me if I think about what the theory definition of conditional probability is.
00:52:24.600 --> 00:52:44.300
Remember, we said the probability of let us say A given B is the probability of A and B over the probability B.
00:52:44.300 --> 00:53:03.500
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
00:53:03.500 --> 00:53:14.900
as well as we know probability of how many females there are, 40% and we also know the and statements.
00:53:14.900 --> 00:53:33.200
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.
00:53:33.200 --> 00:53:38.300
We do not know the probability of A and B.
00:53:38.300 --> 00:53:47.500
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.
00:53:47.500 --> 00:53:49.700
I just care if they voted yes.
00:53:49.700 --> 00:53:56.100
I just want to know that probability and that is going to be the total probability.
00:53:56.100 --> 00:54:11.400
If I want to get that I want to know the probability of female and yes added to the probability of male and yes.
00:54:11.400 --> 00:54:15.600
That is going to give me the probability of yes.
00:54:15.600 --> 00:54:28.400
Although I cannot figure out, I cannot figure out just the probability of yes I could break it down into these two component parts.
00:54:28.400 --> 00:54:35.100
What I am going to do is I am going to use the multiplicative rule in order to find each of these things.
00:54:35.100 --> 00:54:56.200
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.
00:54:56.200 --> 00:55:04.800
We know that this is 40% and the probability of yes given female is 16%.
00:55:04.800 --> 00:55:21.400
This is 60.16 × .4 and that is .064%.
00:55:21.400 --> 00:55:24.700
Let us do the same thing with males.
00:55:24.700 --> 00:55:27.600
Let me draw the line here.
00:55:27.600 --> 00:55:44.800
Let us find the probability of male and yes that would be the probability of yes given male times the probability of male
00:55:44.800 --> 00:56:05.000
and so that would be 7% × 60 there is lots more man, but few percentage of them are voting for this female candidate.
00:56:05.000 --> 00:56:11.800
And that is .042 about 4%.
00:56:11.800 --> 00:56:28.400
If I add .064 and .042 I would get 10.6%.
00:56:28.400 --> 00:56:40.900
What is the probability of a yes 10.6%.
00:56:40.900 --> 00:56:55.500
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,
00:56:55.500 --> 00:57:05.800
so you could think about waiting this with 40% and waiting this with 60% and that is the roughly the same idea.
00:57:05.800 --> 00:57:13.600
That is it for conditional probability for now.
00:57:13.600 --> 00:57:17.700
I will come back to it when we talk about independent events.
00:57:17.700 --> 00:57:19.000
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