A Proof of Bonferroni's Inequality

Bonferroni’s Inequality:

(1) P(A∩B)  ≥ P(A) + P(B) − 1

Proof:

By definition, for all probabilities, P, 0 ≤ P ≤ 1. So, the probability of the union of two events A and B:

(2) P(A∪B)  ≤ 1

Further, we have previously shown:

(3) P(A∪B)  = P(A) + P(B) − P(A∩B)

Substituting RHS of (3) for the LHS of (2) yields:

(4) P(A) + P(B) − P(A∩B)  ≤ 1

Re-arranging terms in (4), yields:

(5) P(A∩B)  ≤ P(A) + P(B) − 1

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