Mutual Independence and Pairwise Independence:
Three events A, B, C are said to be mutually independent if,
P(A∩B) = P(A)×P(B), P(A ∩ C) = P(A)×P(C), P(B ∩ C) = P(B)×P(C)
and P(A∩B∩C) = P(A)×P(B)×P(C)
These events would be said to be pairwise independent if,
P(A∩B) = P(A)×P(B), P(B∩C) = P(B)×P(C) and P(A∩C) = P(A)×P(C).
Thus mutually independent events are pairwise independent but the converse may not be true.
Example -: An event A1 can happen with probability p1 and event A2 can happen with probability p2.
What is the probability that
(i) exactly one of them happens
(ii) at least one of them happens(Given A1 and A2 are independent events).
Sol: (i) The probability that A1 happens is p1
∴ The probability that A1 fails is 1 – p1
Also the probability that A2 happens is p2
Now, the chance that A1 happens and A2 fails is p1(1 – p2) and the chance that A1 fails and A2 happens is p2(1-p1)
∴ The probability that one and only one of them happens is
p1(1 – p2) + p2(1 – p1) = p1 + p2 – 2p1p2
(ii) The probability that both of them fail = (1 – p1) (1 – p2).
∴ probability that atleast one happens=1 -(1 – p1) (1 – p2)=p1+p2 – p1 p2
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