Consider the following Venn diagram. Calculate the value of the probability of 𝐵 given 𝐴.
So the first thing we need to do is look at some mathematical notation. So we’ve got 𝑃, which means the probability, and then our parentheses and then 𝐵 and a vertical line, and then 𝐴. And what this means is the probability that 𝐵 occurs, given that 𝐴 occurs. And it’s this vertical line that means given. So now, what I’ve done is I’ve shrunk the area that we’re gonna look at. So I’ve colored in this whole area orange. And we can disregard this because we’re looking to find the probability that 𝐵 occurs, given that 𝐴 has occurred. So therefore, we know that 𝐴 has to have occurred. And 𝐴 is everything within the left-hand circle. So therefore, we can disregard everything else.
So to solve this problem, we’ve got a couple of methods that we can utilize, one that has a formula that I’m gonna show you in a bit. But first of all, I’m gonna show you a slightly different method. So first of all, we look at our Venn diagram and the values within it. So we’ve got our area of 𝐴. So we’ve got three tenths represents the probability of 𝐴. And that’s because that’s just the section that is only 𝐴. So it doesn’t include the intersection or overlapping part of our Venn diagram. Whereas two tenths represents the area that is the overlapping area or the intersection. And this is where 𝐴 and 𝐵 occur.
In the question, we’re asked to calculate the value of the probability of 𝐵 occurring, given that 𝐴 has occurred. So therefore, we’re gonna pay attention to two tenths because this is the only section where 𝐵 will occur. Now, there’s a couple of ways to use this information to solve the problem. The first way is just to consider our numerators. So we’ve got three and two. And that’s because our denominators are the same. So we can just consider that we’ve got three parts and two parts. So therefore, we can say that our total parts are three plus two which is equal to five. So we can say that total parts within 𝐴 is five.
So therefore, we can say that the probability of 𝐵 occurring given that 𝐴 occurs is equal to two-fifths. And that’s because our two parts represented the part, which was 𝐴 and 𝐵. So it’s where 𝐵 occurred. So therefore, it’s two out of the total five parts. So it’s two-fifths. Another way we can think about it was two tenths out of five tenths. And when we say “out of,” what we mean this context is divided by. So it’s two tenths divided by five tenths. So now, if we divide fractions, what we do is we actually multiply the fractions and we flip the second one or find the reciprocal. So what we have is two tenths multiplied by 10 over five.
So now, to multiply our fractions, what we could do is multiply our numerators then multiply our denominators. So that will give us 20 over 50. However, we don’t need to do that because we can cancel beforehand because we’ve got a 10 on the numerator and a 10 on the denominator. So these will cancel out. So therefore, we would be left with two over five or two-fifths, which is the same answer we got with the other method. So we know that this is correct.
Now, I said that we’re gonna have a couple of methods. So this is one method, but a couple of ways of doing the calculation. But I’m also going to show you a formula that we could have used to get us to the right answer. And that formula is the probability of 𝐵 occurring given that 𝐴 occurs is equal to the probability of 𝐵 and 𝐴 occurring divided by the probability of 𝐴. And if we’d used that formula, that would have taken us to the step that I’ve got up here, which is two tenths divided by five tenths. And that’s because two tenths is the probability of 𝐵 and 𝐴 occurring. And five tenths is the probability of 𝐴 occurring. So it’s gonna take us straight to this step, which we could have then solved to give us our two-fifths.