INDEPENDENT EVENTS

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Independent Events

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◇Overview

Two events $A$ and $B$ are statistically independent if the occurrence (or non-occurrence) of one provides zero information about the probability of the other. Formally, $A$ and $B$ are independent if and only if their joint probability equals the product of their marginal probabilities:
$P(A \cap B) = P(A) \cdot P(B)$

◇Key Definitions

• Product Criterion for Independence: $P(A \cap B) = P(A)P(B)$ (valid even if $P(A)=0$ or $P(B)=0$).
• Conditional Criterion: When $P(B) > 0$, $P(A \mid B) = P(A)$; when $P(A) > 0$, $P(B \mid A) = P(B)$.
• Complement Preservation: If $A$ and $B$ are independent, then $(A, B^c)$, $(A^c, B)$, and $(A^c, B^c)$ are also independent pairs.
• Mutual Independence ($3$ Events): $A, B, C$ are mutually independent iff all pairwise products hold AND $P(A \cap B \cap C) = P(A)P(B)P(C)$.

◇Worked Example

Problem: Two independent servers $S_1$ and $S_2$ have uptime probabilities $P(S_1) = 0.90$ and $P(S_2) = 0.95$. Find (a) the probability that both are up, and (b) the probability that at least one is up.

Solution:

• (a) $P(S_1 \cap S_2) = P(S_1)P(S_2) = 0.90 \times 0.95 = 0.855$.
• (b) $P(S_1 \cup S_2) = 1 - P(S_1^c \cap S_2^c) = 1 - P(S_1^c)P(S_2^c) = 1 - (0.10)(0.05) = 1 - 0.005 = 0.995$.

◇References & Navigation

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• Related Notes: ◇Multiplication Rule, ◇Partition of Sample Space