**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 A_{1} can happen with probability p_{1} and event A_{2} can happen with probability p_{2}.

What is the probability that

(i) exactly one of them happens

(ii) at least one of them happens(Given A_{1} and A_{2} are independent events).

**Sol: ** (i) The probability that A_{1} happens is p_{1}

∴ The probability that A_{1} fails is 1 – p_{1}

Also the probability that A_{2} happens is p_{2}

Now, the chance that A_{1} happens and A_{2} fails is p_{1}(1 – p_{2}) and the chance that A_{1} fails and A_{2} happens is p_{2}(1-p_{1})

∴ The probability that one and only one of them happens is

p_{1}(1 – p_{2}) + p_{2}(1 – p_{1}) = p_{1} + p_{2} – 2p_{1}p_{2}

(ii) The probability that both of them fail = (1 – p_{1}) (1 – p_{2}).

∴ probability that atleast one happens=1 -(1 – p_{1}) (1 – p_{2})=p_{1}+p_{2} – p_{1} p_{2}

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