COMPLEMENTARY RULE

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Complementary Rule

---

◇Overview

The Complementary Rule states that the sum of the probabilities of an event $A$ and its complement $A^c$ is always $1$: $P(A) + P(A^c) = 1$, or equivalently $P(A) = 1 - P(A^c)$. It is derived directly from Kolmogorov's normalization and additivity axioms.

◇Key Definitions

• Complementary Probability Formula: $P(A^c) = 1 - P(A)$ and $P(A) = 1 - P(A^c)$.
• Axiomatic Proof: Since $S = A \cup A^c$ and $A \cap A^c = \emptyset$, Axiom 2 and Axiom 3 give $1 = P(S) = P(A \cup A^c) = P(A) + P(A^c)$.
• "At Least One" Shortcut: For $n$ independent trials with failure probability $q = 1 - p$, $P(\text{at least one success}) = 1 - q^n$.

◇Worked Example

Problem: A packet transmission across a noisy wireless link fails independently with probability $0.10$. If $4$ packets are transmitted, what is the probability that at least one packet fails?

Solution:

• Let $A$ be the event "at least one packet fails." Then $A^c$ is "no packets fail (all 4 succeed)."
• $P(A^c) = (1 - 0.10)^4 = (0.90)^4 = 0.6561$.
• By the Complementary Rule: $P(A) = 1 - P(A^c) = 1 - 0.6561 = 0.3439$.

◇References & Navigation

• Previous Topic: ◇Additive Rule
• Next Topic: ◇Conditional Probability and Multiplicative Rules
• Related Notes: ◇Conditional Probability, ◇Additive and Complementary Rules