# Video: EG17S1-STATISTICS-Q08

EG17S1-STATISTICS-Q08

03:40

### Video Transcript

If 𝐴 and 𝐵 are two events occurring within the sample space 𝑆 of a random experiment, where the probability of 𝐴 is 0.45, the probability of 𝐵 equals 0.6, and the probability of 𝐵 given 𝐴 equals 0.8. Find the probability of 𝐴 given 𝐵.

In our sample space, we have event 𝐴 and event 𝐵. This overlapping space is the intersection of event 𝐴 and event 𝐵. The probability of event 𝐴 occurring is 0.45, which is represented by the yellow oval. It includes the probability of 𝐴 and not 𝐵 and the probability of 𝐴 and 𝐵 both occurring. The probability of 𝐵 occurring is 0.6, the pink oval, which is the probability of only 𝐵 and the probability of 𝐴 and 𝐵 together. We’re also given this probability, the probability of 𝐵 given 𝐴, which is 0.8.

Remember that the probability of an event is the desired outcome over all possible outcomes. The probability of 𝐵 given 𝐴, we know that 𝐴 has already occurred. And if 𝐴 already occurred, then all possible outcomes must come from the yellow circle, the probability of 𝐴. The desired outcome is 𝐵. Since we know that 𝐴 has already happened, the only way for 𝐵 to occur is if it’s in the intersection of 𝐴 and 𝐵. We say that the probability of 𝐵 occurring given 𝐴 must be equal to the probability of the intersection of 𝐴 and 𝐵 over the probability of 𝐴. And by extension, we could say that the probability of 𝐴 given 𝐵 equals the probability of the intersection of 𝐴 and 𝐵 over the probability of 𝐵.

Our end goal is to find the probability of 𝐴 given 𝐵. But since we don’t know the probability of the intersection of 𝐴 and 𝐵, we can’t solve for this yet. We can use this first equation to help us find the probability of the intersection of 𝐴 and 𝐵. We know the probability of 𝐵 given 𝐴 equals 0.8. We’re trying to find the probability of the intersection of 𝐴 and 𝐵. But we know the probability of 𝐴 equals 0.45. To find the probability of the intersection of 𝐴 and 𝐵, we multiply both sides by 0.45. 0.45 times 0.8 equals 0.36. On the right side, 0.45 divided by 0.45 cancels out. And that means that the probability of the intersection of 𝐴 and 𝐵 equals 0.36.

Now that we know the probability of the intersection of 𝐴 and 𝐵, we can find the probability of 𝐴 given 𝐵. The equation for the probability of 𝐴 given 𝐵 is equal to 0.36 divided by 0.6. 0.36 divided by 0.6 equals 0.6. The probability of 𝐴 given 𝐵 equals 0.6.