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The probability of A given B and the probability of B given A are generally different because they condition on different events. Bayes' theorem connects these two conditional probabilities by incorporating the prior probabilities of the individual events, $P(A)$ and $P(B)$.

Conditions: A and B are events in a probability space.; $P(B) > 0$ and $P(A) > 0$.

The probability of A given B and the probability of B given A are generally different because they condition on different events. Bayes' theorem connects these two conditional probabilities by incorporating the prior probabilities of the individual events, $P(A)$ and $P(B)$.

Conditions: A and B are events in a probability space.; $P(B) > 0$ and $P(A) > 0$.