ADDITIVE RULE

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

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◇Overview

The Additive Rule of Probability (also known as the Principle of Inclusion-Exclusion for two events) calculates the probability that at least one of two events $A$ or $B$ occurs, denoted $P(A \cup B)$. Because summing $P(A)$ and $P(B)$ counts outcomes in the overlap $A \cap B$ twice, the intersection probability $P(A \cap B)$ must be subtracted once.

◇Key Definitions

• Union ($A \cup B$): The event that $A$ occurs, $B$ occurs, or both occur simultaneously.
• General Additive Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ for any two events $A$ and $B$.
• Special Additive Rule (Mutually Exclusive Events): When $A \cap B = \emptyset$, $P(A \cap B) = 0$, simplifying the rule to $P(A \cup B) = P(A) + P(B)$.
• Three-Event Inclusion-Exclusion: $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$.

◇Worked Example

Problem: In a university server cluster, $P(\text{High CPU}) = 0.35$, $P(\text{High Memory}) = 0.25$, and the probability of both occurring simultaneously is $P(\text{High CPU} \cap \text{High Memory}) = 0.12$. What is the probability that a server experiences at least one of these resource spikes?

Solution:
$P(\text{CPU} \cup \text{Mem}) = 0.35 + 0.25 - 0.12 = 0.48$

◇References & Navigation

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