Module 2 · Probability Foundations · Topic 5

Multiplication Rule

Independent Events. If A and B are independent (one doesn’t affect the other), then P(B | A) = P(B), and the rule simplifies to P(A ∩ B) = P(A) × P(B).

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Key formulas

General Multiplication Rule

P(A ∩ B) = P(A) × P(B | A)

From the Probability Foundations formula sheet

Conditional Probability: P(A | B) = P(A and B) / P(B) — Probability of A after restricting attention to cases where B occurred.
Bayes' Theorem: P(A | B) = P(B | A)P(A) / P(B) — Updates a prior probability after observing evidence.

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Worked example

With Replacement

Drawing two cards with replacement: P(both aces) = P(ace) × P(ace) = (4/52)(4/52) = 16/2704 ≈ 0.0059.

Without Replacement

Drawing two cards without replacement: P(both aces) = P(1st ace) × P(2nd ace | 1st ace) = (4/52)(3/51) = 12/2652 ≈ 0.0045.

Related glossary terms

Independence: Two events are independent when one occurring does not change the probability of the other. Example: Two separate fair coin flips are independent.

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