INTERSECTION

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇intersection

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◇Overview

The Intersection of two events $A$ and $B$, denoted $A \cap B$ (or $AB$), is the event consisting of all outcomes $\omega \in S$ that belong to both $A$ and $B$ simultaneously (logical AND). In words, $A \cap B$ occurs on a trial if and only if both $A$ and $B$ occur on that trial.

◇Key Definitions

• Set Definition: $A \cap B = \{\omega \in S : \omega \in A \text{ and } \omega \in B\}$.
• Probability Bounds (Fréchet Bounds): Since $A \cap B \subseteq A$ and $A \cap B \subseteq B$:
  $\max(0, P(A) + P(B) - 1) \le P(A \cap B) \le \min(P(A), P(B))$
• Three Ways to Compute $P(A \cap B)$:
  1. From Union (Additive Rule Rearranged): $P(A \cap B) = P(A) + P(B) - P(A \cup B)$.
  2. From Conditional Probability (General Multiplicative Rule): $P(A \cap B) = P(A)P(B \mid A) = P(B)P(A \mid B)$.
  3. Under Independence: $P(A \cap B) = P(A)P(B)$.

◇Worked Example

Problem: Suppose $P(A) = 0.70$ and $P(B) = 0.60$. Find the tightest possible lower and upper bounds (Fréchet bounds) on $P(A \cap B)$ without knowing how $A$ and $B$ are related.

Solution:

• Upper Bound: Since $A \cap B \subseteq B$, $P(A \cap B) \le \min(0.70, 0.60) = 0.60$ (achieved when $B \subseteq A$).
• Lower Bound (Bonferroni Inequality): Since $P(A \cup B) \le 1$, $P(A \cap B) = P(A) + P(B) - P(A \cup B) \ge 0.70 + 0.60 - 1.00 = 0.30$.
• Thus $0.30 \le P(A \cap B) \le 0.60$.

◇References & Navigation

• Previous Topic: ◇Union
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• Related Notes: ◇Mutually Exclusive Events, ◇Types of Event Operations