Video: Using Bayes’ Rule to Find the Conditional Probability of an Event

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴) = 0.63 and 𝑃(𝐡) = 0.77. Given that 𝑃(𝐡|𝐴) = 0.88, find 𝑃(𝐴|𝐡).

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Video Transcript

Suppose that 𝐴 and 𝐡 are events with probabilities: probability of 𝐴 is 0.63 and probability of 𝐡 is 0.77. Given that the probability of 𝐡 given 𝐴 is equal to 0.88, find the probability of 𝐴 given 𝐡.

We can find the probability of 𝐴 given 𝐡 using a formula. We can use the formula the probability of 𝐴 given 𝐡 is equal to the probability of 𝐡 given 𝐴 times the probability of 𝐴 divided by the probability of 𝐡. Or we could use the formula the probability of 𝐴 given 𝐡 is equal to the probability of 𝐴 and 𝐡 divided by the probability of 𝐡.

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

But looking at what we’re given, we are given the probability of 𝐴, the probability of 𝐡, and the probability of 𝐡 given that 𝐴 has already happened. And all three of those are found here in this formula. So this is what we’ll use.

The probability of 𝐡 given 𝐴 is 0.88. The probability of 𝐴 is 0.63. And the probability of 𝐡 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 𝐴 happening given that 𝐡 has already happened is 0.72.

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