Is the following statement true? Pairwise independent events are mutually independent

No. Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree Each pair is independent, but P(A∩B∩C) = 1/4 ≠ 1/8 = P(A)P(B)P(C).

Counterexample

Is this true?

Pairwise independent events are mutually independent

No — it is false.

The counterexample

Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree

Why it fails

Each pair is independent, but P(A∩B∩C) = 1/4 ≠ 1/8 = P(A)P(B)P(C).

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Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Convergence in probability ⇒ almost sure convergence
Uncorrelated ⇒ independent
Every random variable has a moment generating function

See real exam options built on this mistake Drill statements like this

probability

Go deeper· Explained in full in Probability axioms, conditioning and independence· Asked in CSIR NET December 2025· From Probability › Axioms, conditional probability, independence, Bayes

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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