Let A1, A2,, An be an arbitrary collection of n events in a sample space . Show

Let A1, A2,…, An be an arbitrary collection of n events in a sample space Ω. Show that

This is known as Bonferroni’s inequality.
(Hint: Apply Boole’s inequality from the last exercise to the events A′i , i = 1, 2,…, n.)

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n n P(AAA)1- P(A) = P(A;)-(n 1).