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The denominator $P(B)$ ensures that the posterior probability $P(A \mid B)$ sums to 1 over all possible mutually exclusive events $A$. It scales the product of the likelihood and prior to a valid probability distribution.

Conditions: Bayes' theorem is written in probability form: $P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}$.

The denominator $P(B)$ ensures that the posterior probability $P(A \mid B)$ sums to 1 over all possible mutually exclusive events $A$. It scales the product of the likelihood and prior to a valid probability distribution.

Conditions: Bayes' theorem is written in probability form: $P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}$.

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.