### Video Transcript

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

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

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

Our end goal is to find the probability of π΄ given π΅. But since we donβt know the probability of the intersection of π΄ and π΅, we canβt solve for this yet. We can use this first equation to help us find the probability of the intersection of π΄ and π΅. We know the probability of π΅ given π΄ equals 0.8. Weβre trying to find the probability of the intersection of π΄ and π΅. But we know the probability of π΄ equals 0.45. To find the probability of the intersection of π΄ and π΅, 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 π΄ and π΅ equals 0.36.

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