Types of Event Operations

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Types of Event Operations

◇Overview

Because events are subsets of the sample space $S$ , Event Operations correspond directly to set-theoretic operations—Union ( $\cup$ ), Intersection ( $\cap$ ), Complement ( $^c$ ), Set Difference ( $\setminus$ ), and Symmetric Difference ( $\triangle$ )—as well as propositional logic connectives (OR, AND, NOT, XOR).

◇Key Definitions

Verbal Event Statement	Set Operation	Probability Formula
At least one of $A$ or $B$	$A \cup B$	$P(A) + P(B) - P(A \cap B)$
Both $A$ and $B$	$A \cap B$	$P(A)P(B \mid A)$
Not $A$	$A^c$	$1 - P(A)$
$A$ occurs, but NOT $B$ ( $A$ only)	$A \setminus B = A \cap B^c$	$P(A) - P(A \cap B)$
Exactly one of $A$ or $B$	$A \triangle B = (A \cap B^c) \cup (A^c \cap B)$	$P(A) + P(B) - 2P(A \cap B)$
Neither $A$ nor $B$	$(A \cup B)^c = A^c \cap B^c$	$1 - P(A \cup B)$

Algebraic Laws: Commutative, Associative, Distributive ( $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ ), and De Morgan's Laws ( $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$ ).

◇Worked Example

Problem: Given $P(A) = 0.50$ , $P(B) = 0.40$ , and $P(A \cap B) = 0.15$ , find:

$P(\text{A only}) = P(A \cap B^c) = P(A) - P(A \cap B) = 0.50 - 0.15 = 0.35$ .
$P(\text{Exactly one of A or B}) = P(A) + P(B) - 2P(A \cap B) = 0.50 + 0.40 - 2(0.15) = 0.60$ .

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Related Notes: ◇intersection, ◇Sample Space

◇Overview

◇Key Definitions

◇Worked Example

◇References & Navigation

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