MULTIPLICATIVE RULE

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

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◇Overview

In probability theory, the Multiplicative Rule of Probability determines the probability of the intersection of two or more events, $P(A \cap B)$. It comes in two forms: the General Multiplicative Rule (for any events, using conditional probability) and the Special Multiplicative Rule (for independent events).

◇Key Definitions

• General Multiplicative Rule (Dependent or Independent Events):
  $P(A \cap B) = P(A) P(B \mid A) = P(B) P(A \mid B)$
• Special Multiplicative Rule (Independent Events Only): If and only if $A$ and $B$ are independent:
  $P(A \cap B) = P(A) P(B)$
• Extension to $k$ Independent Events: If $A_1, A_2, \dots, A_k$ are mutually independent, $P(A_1 \cap A_2 \cap \cdots \cap A_k) = \prod_{i=1}^k P(A_i)$.

◇Worked Example

Problem: Contrast drawing $2$ Aces from a $52$-card deck (a) with replacement vs. (b) without replacement.

Solution:

• (a) With Replacement (Independent): Special Multiplicative Rule applies: $P(A_1 \cap A_2) = P(A_1)P(A_2) = \frac{4}{52} \times \frac{4}{52} = \frac{1}{169} \approx 0.00592$.
• (b) Without Replacement (Dependent): General Multiplicative Rule applies: $P(A_1 \cap A_2) = P(A_1)P(A_2 \mid A_1) = \frac{4}{52} \times \frac{3}{51} = \frac{1}{221} \approx 0.00452$.

◇References & Navigation

• Previous Topic: ◇Conditional Probability
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• Related Notes: ◇Generalized Multiplication, ◇Conditional Probability and Multiplicative Rules