Probability Multiplication Rules

The probability multiplication rules give the probability of the intersection of two events in terms of a conditional probability.

P(A \cap B) = P(A)\,P(B \mid A)

= P(B)\,P(A \mid B)

• This rearranges the definition of conditional probability, P(B \mid A) = P(A \cap B)/P(A).
• For independent events it reduces to P(A \cap B) = P(A)\,P(B).
• It extends to three events as P(A \cap B \cap C) = P(A)\,P(B \mid A)\,P(C \mid A \cap B).

Drawing Two Red Marbles Without Replacement A bag has 5 red and 3 blue marbles. Two are drawn without replacement. What is the probability both are red?

Answer

\begin{align*} P(R_1 \cap R_2) &= P(R_1)\,P(R_2 \mid R_1) \\ &= \frac{5}{8} \cdot \frac{4}{7} \\ &= \frac{20}{56} = \frac{5}{14} \approx 0.357 \end{align*}