Multiplication Theorem and Independent Events

Rearranging the definition of conditional probability gives a rule for the probability that both events occur — the probability of the intersection $A \cap B$.

This is the natural tool for "and then" problems, especially drawing objects one after another. Whether the drawing is with or without replacement matters: without replacement the second draw is conditional on the first (the contents have changed), so the conditional factor is genuinely different from the plain probability.

Independent events. Two events are independent when the occurrence of one does not affect the probability of the other, i.e. $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$. Substituting into the multiplication theorem gives the clean test and rule: