What is Bayes’s theorem for conditional probability? We'll be proving it briefly in today's lesson, as it is a direct result of the definition of conditional probabilities. We'll also see how Baye's theorem works with a sample space partitioned into separate events, and an example of applying Baye's theorem! Baye's theorem gives us a way to calculate the conditional probability P(A|B) using P(B|A).

We know that the probability of A given B, P(A|B), is equal to P(A and B)/P(B) and P(B|A) = P(B and A)/P(A) = P(A and B)/P(A). Then, multiplying both sides of the second equation by P(A) we have that P(B|A)*P(A) = P(A and B) and we can substitute this into our expression for P(A|B). Thus, we have P(A|B) = P(B|A)P(A)/P(B), and this is Baye's theorem (sometimes called Baye's rule). Often times, we will have to calculate P(B) in the denominator using the Law of Total Probability.

I meant to put this in the video as a caption: the example problem is from "Probability" by Jim Pitman.

Intro to Conditional Probability:    • Intro to Conditional Probability | Pr...  
Law of Total Probability:    • The Law of Total Probability | Probab...  

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