ADDITIVE AND COMPLEMENTARY RULES

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Additive and Complementary Rules

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◇Overview

Combining the Additive Rule and the Complementary Rule allows efficient computation of complex event probabilities—especially statements involving "at least one", "neither", or "at most one"—often bypassing tedious case-by-case enumeration.

◇Key Definitions

• Complementary Rule: For any event $E$, $P(E^c) = 1 - P(E)$, since $E \cup E^c = S$ and $E \cap E^c = \emptyset$.
• De Morgan's First Law (Neither $A$ nor $B$): $(A \cup B)^c = A^c \cap B^c$, so $P(A^c \cap B^c) = 1 - P(A \cup B)$.
• De Morgan's Second Law (Not both $A$ and $B$): $(A \cap B)^c = A^c \cup B^c$, so $P(A^c \cup B^c) = 1 - P(A \cap B)$.

◇Worked Example

Problem: Suppose $P(A) = 0.50$, $P(B) = 0.40$, and $P(A \cap B) = 0.20$. Find the probability that neither $A$ nor $B$ occurs, $P(A^c \cap B^c)$.

Solution:

1. First apply the Additive Rule: $P(A \cup B) = 0.50 + 0.40 - 0.20 = 0.70$.
2. By De Morgan's Law and the Complementary Rule: $P(A^c \cap B^c) = P((A \cup B)^c) = 1 - P(A \cup B) = 1 - 0.70 = 0.30$.

◇References & Navigation

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• Related Notes: ◇Relative Frequency Form, ◇Complementary Rule