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Bayes' theorem is derived by expressing the same intersection $P(A \cap B)$ in two different ways using the definition of conditional probability. First, $P(A \mid B) = P(A \cap B) / P(B)$ implies $P(A \cap B) = P(A \mid B)P(B)$.

Conditions: Events A and B have positive probability ($P(A) > 0, P(B) > 0$).; The definition of conditional probability applies.

Bayes' theorem is derived by expressing the same intersection $P(A \cap B)$ in two different ways using the definition of conditional probability. First, $P(A \mid B) = P(A \cap B) / P(B)$ implies $P(A \cap B) = P(A \mid B)P(B)$.

Conditions: Events A and B have positive probability ($P(A) > 0, P(B) > 0$).; The definition of conditional probability applies.

Bayes' theorem is derived by expressing the joint probability $P(A \cap B)$ in two different ways using the definition of conditional probability. First, $P(A \mid B) = P(A \cap B) / P(B)$ implies $P(A \cap B) = P(A \mid B)P(B)$.

Conditions: $P(B) > 0$; $P(A) > 0$

Bayes' theorem is derived by expressing the joint probability $P(A \cap B)$ in two different ways using the definition of conditional probability. First, $P(A \mid B) = P(A \cap B) / P(B)$ implies $P(A \cap B) = P(A \mid B)P(B)$.

Conditions: $P(B) > 0$; $P(A) > 0$