Probability Multiplication Rule

Study guide: Exam P-1

The Probability Multiplication Rule gives the joint probability of two events A and B occurring together.

• When A and B are independent, the rule simplifies to P(A \cap B) = P(A) \cdot P(B) since P(B \mid A) = P(B)
• For a chain of events, the rule extends to P(A_1 \cap A_2 \cap \cdots \cap A_n) = P(A_1)\,P(A_2 \mid A_1)\,P(A_3 \mid A_1 \cap A_2) \cdots

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

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

Sequential Claim Filing Without Replacement A portfolio has 10 policies: 4 will generate claims this year and 6 will not. Two policies are selected at random without replacement. What is the probability both generate claims?

Answer Let A = first policy generates a claim, B = second policy generates a claim. Applying the multiplication rule: P(A \cap B) = P(A) \cdot P(B \mid A) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15} \approx 0.133