Let `A` ,`B` ,`C` be three events. If the probability of occurring exactly one event out of `A` and `B` is `1-x `, out of `B` and `C` is `1-2x `, out of `C` and `A` is `1-x ` and that of occurring three events simultaneously is `x^2` , then prove that the probability that at least one out of `A`, `B`, `C` will occur is greater than 1/2 .

Explanation

`=P[(A cap B^{prime}) cup(A^{prime} cap B)] `

`=P(A cup B)-P(A cap B) `

`=P(A)+P(B)-2 P(A cap B) `

`therefore P(A)+P(B)-2 P(A cap B) `

`quad=1-a`

Similarly,

`P(B)+P(C)-2 P(B cap C) `

`=1-2 a quad ldots . .(2) `

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