Starting from the equality P(A)P(B|A) = P(B)P(A|B), you can solve for either conditional probability by dividing by the corresponding marginal probability. Dividing both sides by P(B) isolates P(A|B), giving P(A|B) = P(A)P(B|A)/P(B).
Conditions: Both A and B have positive probability for the ordinary conditionals used here.; The algebraic rearrangement requires the denominators P(B) and P(A) to be nonzero.
Starting from the equality P(A)P(B|A) = P(B)P(A|B), you can solve for either conditional probability by dividing by the corresponding marginal probability. Dividing both sides by P(B) isolates P(A|B), giving P(A|B) = P(A)P(B|A)/P(B).
Conditions: Both A and B have positive probability for the ordinary conditionals used here.; The algebraic rearrangement requires the denominators P(B) and P(A) to be nonzero.
Bayes' theorem is true because the joint probability of two events can be decomposed in two symmetric ways. The probability that both A and B occur is equal to the probability of A multiplied by the conditional probability of B given A, and also equal to the probability of B multiplied by the conditional probability of A given B.
Conditions: A and B are events in the same probability space.; Both A and B have positive probability for the ordinary conditionals used here.
Bayes' theorem is true because the joint probability of two events can be decomposed in two symmetric ways. The probability that both A and B occur is equal to the probability of A multiplied by the conditional probability of B given A, and also equal to the probability of B multiplied by the conditional probability of A given B.
Conditions: A and B are events in the same probability space.; Both A and B have positive probability for the ordinary conditionals used here.