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Question

Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are:

(i) p1p2 (ii) (1 - p1)p2 (iii) 1 - (1 - p1)(1 - p2) (iv) p1 + p2 - 2p1p2 [NCERT EXEMPLAR]

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Solution

i As, p1p2=PA×PBAnd, A and B are independent events.i.e. PA×PB=PABSo, PAB=p1p2Hence, p1p2=PA and B occurii As, 1-p1p2=1-PA×PB=PA×PBAnd, A and B are independent events.i.e. PA×PB=PABSo, PAB=1-p1p2Hence, 1-p1p2=P A does not occur, but B occursiii As, 1-1-p11-p2=1-1-PA×1-PB=1-PA×PBAnd, A and B are independent events.i.e. PA×PB=PAB1-1-p11-p2=1-PAB=1-PAB=PABSo, PAB=1-1-p11-p2Hence, 1-1-p11-p2=PAt least one of A and B occursiv As, p1+p2-2p1p2=p1-p1p2+p2-p1p2=PA-PA×PB+PB-PA×PBAnd, A and B are independent events.i.e. PA×PB=PABp1+p2-2p1p2=PA-PAB+PB-PAB=Ponly A+Ponly BSo, Ponly A+Ponly B=p1+p2-2p1p2Hence, p1+p2-2p1p2=PExactly one of A and B occurs

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