# Video: Using Bayesβ Rule to Find the Conditional Probability of an Event

Suppose that π΄ and π΅ are events with probabilities π(π΄) = 0.63 and π(π΅) = 0.77. Given that π(π΅|π΄) = 0.88, find π(π΄|π΅).

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### Video Transcript

Suppose that π΄ and π΅ are events with probabilities: probability of π΄ is 0.63 and probability of π΅ is 0.77. Given that the probability of π΅ given π΄ is equal to 0.88, find the probability of π΄ given π΅.

We can find the probability of π΄ given π΅ using a formula. We can use the formula the probability of π΄ given π΅ is equal to the probability of π΅ given π΄ times the probability of π΄ divided by the probability of π΅. Or we could use the formula the probability of π΄ given π΅ is equal to the probability of π΄ and π΅ divided by the probability of π΅.

Notice, however, in this second formula, we have the probability of π΄ and π΅. In a sample space, here will be event π΄ and here will be event π΅. Where π΄ and π΅ overlap would be the intersection of π΄ and π΅. So the probability of π΄ and π΅ would be inside of here.

But looking at what weβre given, we are given the probability of π΄, the probability of π΅, and the probability of π΅ given that π΄ has already happened. And all three of those are found here in this formula. So this is what weβll use.

The probability of π΅ given π΄ is 0.88. The probability of π΄ is 0.63. And the probability of π΅ is 0.77. Multiplying on the numerator, we get 0.5544. And now we need to divide by 0.77. And we get 0.72. So this means that the probability of π΄ happening given that π΅ has already happened is 0.72.