 Hello, in this video we will discuss conditional probabilities. So I have pictured on the screen various shapes of various sizes of various colors. Now if I asked you what was the probability of getting a yellow triangle, you see I have 16 different shapes of varying colors and there's only one that's a yellow triangle so the probability would be 1 out of 16. Now what if I told you find the probability that you get a triangle given that the shape is yellow? That's what a conditional probability is, that means you're literally just going to take your sample space and restrict it to the condition, in this case that means yellow shape, I said given that the shape is yellow and then you'll calculate the probability that the shape is a triangle. Well out of the four yellow shapes, only one is a triangle, so 1 out of 4. So the official definition is the conditional probability is the likelihood that an event will occur given that another event has already occurred. So we've seen this notation a little bit, you see probability of A vertical line B, that means find the probability of event A occurring given that event B has already occurred. So the formal formula for the probability of A occurring given that the condition that B has occurred is the probability of A and B occurring divided by the probability of B and the other formula is the probability of event B occurring given that event A has occurred is the probability of A and B occurring divided by the probability of A occurring. And unless you take some sort of fancy theoretical stats class later on in the future during your future degree program, you really won't need these formulas. The intuitive approach is the conditional probability of B given A can be found by assuming that event A has occurred and then calculating the probability that event B will occur. So basically you're going to shrink the sample space, that's what conditional probability is doing. You shrink the sample space based on your condition, based on the given that condition. Shrink the sample space, I can't emphasize that enough. So there is a difference in calculating the probability of A occurring given that B occurred and finding the probability that B occurs given that A has occurred. They're both two separate events, so the fact that one occurs before the other does make a difference. Incorrectly mixing these up is called confusion of the inverse, I just wanted to tell you there is a difference. So refer to the table to find the probability that a subject actually uses drugs given that he or she had a positive test result. So given that he or she had a positive test result, that's your condition, you need to shrink the sample space. That means you only want to look at positive drug tests, so everything else is obsolete, all the other numbers are obsolete, they don't matter for this question. Your condition, your sample size is reduced to positive test results only. And calculating the probability that a subject uses drugs, given that, the vertical bar for given that, they have a positive test result. So there's what, 50 positive test results, so out of the 50 positive test results, remember that's my newly refined sample space, how many use drugs? Well that would be the 41, 41 out of 50, also known as .82. So the key is to shrink that sample space down. A couple wishes to have four children, find the probability a couple will have a boy as their fourth child, given that, there's that word for conditional probability, given that the first three are boys. Is it the same as getting all four boys? Assume boys and girls are equally likely. So I want to calculate the probability, the fourth child is a boy, given that the first three are boys. Well when a couple has four kids, my condition is that the first three are boys. So what is my refined sample space? What is my shrunk down sample space? Because there are 16 outcomes total when you have four children. Well the refined sample space is the given that the first three children are boys. So I have the first three children are boys and the fourth child could be a girl. I have the first three children are boys and the fourth child could be a boy. Those are the only items in my refined sample space. There's two items total. Refined sample space, how many are all four boys? Just one. So one out of two or .5. The question is how does this compare to getting all four boys? So without the condition, the probability of getting four boys would be, there would be 16 outcomes total. That's two times two times two times two, four children, two outcomes a piece gives you 16. And only one of those, if you were to list them all, would give you all four boys. So that would be one out of 16. So clearly the two probabilities are not the same. That condition does make a difference. That's why we have conditional probabilities. Well, we have a lady who has the following collection of hardcover and paperback books, both fiction and nonfiction. She randomly selects one of the books to read. Calculate the book, probability the book is hardcover. Given, there's the word given, it is fiction. So my condition here is given that it's fiction. So we need to shrink the sample space down to just fiction books. So we only care about fiction books. Nothing else matters. Don't care about any other numbers, just those that are fiction. So my notation is the probability of book is hardcover. Given that, vertical bar, it is fiction. We restricted the sample space to the 11 fiction books only. That's my sample space. So of those 11 fiction books, how many are hardcover? Just three. So the answer is three out of 11. Or 0.2727. That's the answer to the question. So conditional probability means shrink that sample space down. Make your life easier. That's all I have for now. Thanks for watching.