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Conditional Probability

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Hi. This tutorial covers conditional probability. So let's start with a Venn diagram. So what this Venn diagram shows are basically two characteristics, somebody that has blond hair or somebody that has brown eyes.

So this whole circle here represents those with blond hair. This circle represents those with brown eyes. Remember, the intersection part represents the people with both blond hair and brown eyes. And anything in the outside region that's not in the circle, but within the rectangle, represents those with neither brown eyes nor blond hair.

So for simplicity, let's just define two events here. So let's let event A be the event that a person has blond hair-- so this is A-- and event B be the event that the person has brown eyes. So this is going to be B over here.

So let's just start by determining some probabilities that we're eventually going to use to calculate some conditional probabilities. So let's start with just the probability of A. So A is blond. B is brown. So if we want the probability of blond hair, we want to count the probabilities that represent somebody that will have blond hair.

Now, you might be inclined just to say 12%. But what the 12% means is people that have only blond hair but not brown eyes. So for probability of A, we actually want to count both of these two probabilities, the 12% and the 7%, because both of those two outcomes will include people with blond hair. So what this ends up being is 0.12 plus 0.07. And so this probability is going to be 0.19, so 19% chance of selecting somebody with blond hair.

Now, if we do the probability of B, probability of selecting somebody with brown eyes, we would just need to add up the 0.07 plus the 0.35. And that's going to give me 0.42, so 42% chance of selecting somebody with brown eyes. Now, if we want the probability of A-- remember this symbol is and-- so A and B, so the probability of somebody with blond hair and brown eyes, well, that's simply just this overlapping region. So that's just 7%. So we don't need to do any calculation there. That number is provided for us.

Now the probability of A "or"-- remember the U symbol is "or" or union, so the union of A and B, or A or B. Remember that "or" is an inclusive or. So it also includes this "and" probability. So if we want A or B, all we need to do is simply add up the three probabilities in the circles.

So if we end up adding all of those, that will give me 0.54. And this number should be the complement of this number. And it is. These two add up to 1. So there's a 54% chance that somebody has either blond hair or brown eyes or both. So that shouldn't really be anything new yet.

What is a more interesting question I could try to answer using this data? How about, what is the probability of selecting someone that has blond hair if it is known that he or she has brown eyes? So this is our first conditional probability question. So let's start with some notation just so you can see how to translate this question into symbols.

So the notation here is going to be the probability-- now, what we're going to do is we're going to put in a vertical bar here. And we're going to put two events. We're going to put an event on each side of that vertical bar. This vertical bar is read as "given." So basically what comes after the given is the condition.

And the condition is what we already know. So that comes from this part-- if it is known that he or she has brown eyes. So remember, brown eyes was event B. And we now want to know well, what's the probability of blond hair, given he or she has brown eyes? So this probability is going to be read as the probability of A given B.

So now let's go back and see if we can actually figure out what this probability is. So remember that our given was that the person had brown eyes. So if we look at brown eyes, we want to know, of all of the people with brown eyes-- remember that's the 42%. So out of these 42%, what's the probability then of selecting somebody that also has blond hair? So we're looking at the 7% out of the entire 42% of people that have brown eyes.

So this probability is going to end up being A given B equals the 7%, 0.07, divided by the 42%, so 0.07 divided by 0.42. So if we actually divide those, 0.07 divided by 0.42, we end up with 0.167. So that's about 16.7%. So basically if we translate that back into this picture, about 16.7% of brown-eyed people also have blond hair. So 16.7% of brown-eyed people have blond hair.

So let's translate that now into a formula. So conditional probability is the probability of an event if it is known that another event has occurred. So we are trying to find the probability of blond hair if it's known that person had brown eyes.

So the formula that we used here, we used the probability of A given B. And what it was is the probability of A and B divided by the probability of B. And this is what we did. So what we did is, we took the 0.07, which was that overlapping region, and we divided by the probability of brown eyes, which was 0.42. So we divided those two. And these came pretty much right off of the Venn diagram.

Now, if you switch it around, sometimes you'll have it as the probability of B given A. Notice what changes. The numerator of the fraction stays the same. But the denominator changes to the condition. So you're always dividing by the probability of the condition of what is already known.

Let's do one last example now. So same Venn diagram, same situation. So we want to know, what is the probability of selecting someone that does not have brown eyes if it is known that he or she has blond hair?

So let's just start by notating this probability. Now the given-- it's known that he or she has blond hair. So we know the given condition is blond hair. So that's event A. Now, what we want to know is the probability of someone that does not have brown eyes. So remember that we're looking for the complement of B. So we'll write that as B prime, so the probability of B prime given A.

So using the formula, this is going to be the probability of B prime and A, so the complement of B and A, over the probability of A. So let's do the denominator first. So the probability of A, the probability of someone with blond hair is that 19%, so these two added up, so 0.19.

Now the probability of somebody that does not have brown eyes but does have blond hair, that's just going to be the 12%, because this 12% are people with blond hair but people that don't have brown eyes, because the brown-eyed people are in this region. So it's going to end up being 0.12 divided by 0.19.

So if we do that on the calculator, we end up with about 63.2%, 0.632. So 63.2% of blond-haired people do not have brown eyes. So this is just another example of how to calculate conditional probability using the formula that we already had.

All right. So that's been your tutorial on conditional probability Thanks for watching.