Addition and Multiplication Rules

The addition rule gives you $P(\text{either})$. The multiplication rule gives you $P(\text{both})$. Swap them when evaluating an applicant with two risk factors and you misprice the policy by a factor of five.

For any two events:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Special case, mutually exclusive: $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

Special case, independent: $P(A \cap B) = P(A) P(B)$, so $P(A \cup B) = P(A) + P(B) - P(A) P(B)$.

HIGH-FREQUENCY: The exam often gives you two of the four quantities $P(A)$, $P(B)$, $P(A \cup B)$, $P(A \cap B)$ and asks for the missing one. Rearrange:

$$P(A \cap B) = P(A) + P(B) - P(A \cup B)$$

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

Common mistakes

• Forgetting to subtract $P(A \cap B)$ in the addition rule. With $P(A) = 0.40$, $P(B) = 0.35$, writing $P(A \cup B) = 0.75$ ignores overlap. Trap: 0.75 instead of 0.55.
• Using the simple multiplication rule without replacement. If 8 of 20 marbles are red, the probability of two reds is $(8/20)(7/19) = 0.1474$, not $(8/20)^2 = 0.16$. Trap: 0.16.
• Confusing "or" and "and." "A or B" = union = addition rule. "A and B" = intersection = multiplication rule. Swapping changes the answer entirely.

Bottom line

• Addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$; rearrange to solve for any of the four quantities
• Multiplication rule: $P(A \cap B) = P(A \mid B) \cdot P(B)$; simplifies to $P(A) P(B)$ only under independence
• Chain rule: $P(A \cap B \cap C) = P(A) \cdot P(B \mid A) \cdot P(C \mid A \cap B)$
• Without replacement: update the denominator at each step (20, 19, 18,...)

Exam shortcut

Label every probability "union" or "intersection" before writing any formula. This eliminates the most common error. For without-replacement problems, write out the denominator sequence (20, 19, 18,...) first. "Or = Add minus overlap. And = Multiply." Two rules, one sentence. Feasibility check: $P(A \cap B)$ must be between $\max(0, P(A) + P(B) - 1)$ and $\min(P(A), P(B))$. If your answer violates this, you made an error.

The full lesson (about 2,073 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1e

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