If A, B and C are independent events such that P(A) = P(B) = P(C) = p,

1.	If A, B and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of A, B and C.
Answer» Given, P(A) = P(B) = P(C) p P(occurrence of at least two of A,B,C) = P(A)P(B)P(\(\bar C\)) +P(A)P(\(\bar B\))P(C) + P(\(\bar A\))P(B)P(C) + P(A)P(B)P(C) = p × p(1 - p) + p × (1 - p)p + (1 - p)p × p + p × p × p =3p² (1 - p) + p³ = 3p² -3p³ +p³ = 3p² -2p³

If A, B and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of A, B and C.

Answer»

Given, P(A) = P(B) = P(C) p

P(occurrence of at least two of A,B,C) = P(A)P(B)P(\(\bar C\)) +P(A)P(\(\bar B\))P(C) + P(\(\bar A\))P(B)P(C) + P(A)P(B)P(C)

= p × p(1 - p) + p × (1 - p)p + (1 - p)p × p + p × p × p

=3p² (1 - p) + p³

= 3p² -3p³ +p³

= 3p² -2p³

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If A, B and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of A, B and C.

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