Consider four independent events ?A?1, ?A?2, ?A?3, and ?A?4, and let pi = P(A) for I = 1,2,3,4. Express the probability that at least one of these four events occurs in terms of the ?p?is, and do the same for the probability that at least two of the events occur.

Problem 112E Answer: Step1: We have four independent events A , A , A , A and 1 2 3 4 Let p =i(A) , i =i,2,3,4. We need to Express the probability that at least one of these four events occurs in terms of the p is, and do the same for the probability that at least two of the events occur. Step2: Consider, the probability that none of these events occurs is q = 1 - p = P(A ) i i 1i Let “x” denotes the event that at least one of the A events occurs. i Now, P(x) = 1 - P(x ) 1 = 1 - P(A A 1A A 2 13 14 = 1 - P(A )•P(A )•P(A )•P(A ) 1 1 1 2 3 4 = 1 - q •q •q •q 1 2 3 4 Therefore, the probability that at least one of the A events occurs is P(x) = 1 - q •q •q •q . i 1 2 3 4 Step3: Let Y denote the event that at least two of the A events occurs. i Now, P(Y) = 1 - P(Y ) 1 = 1 - [P(x ) + P(Exactly one A)] i 4 = 1 - P (x ) -[ P(only Ai occurs )] ………(1) i=1 Where, 4 1 1 1 1 1 1 1 1 P(only Ai occurs ) = {[P(A A A A )]+[P(A1 A A2 A ]+[3(A A4A 1 2 3 4 1 2 3 i=1 1 1 1 1 A ]4P(A A A1A ]} 2 3 4 = [(p 1q •q2 •q 3+(p4q •q2•q )1(p 3 4 3 •q 1q •q2 )+(4 •q •4 •q1)] 2 3 Substitute these into above equation (1) we get, 1 P(Y) = 1 - P(x ) - (p •q •q1 •q 2-(p3•q •4 •q 2-(p1•q •3 •q4-(p3•q •1 •q2 ) 4 4 1 2 3