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Let A ,B ,C be three events. If the prob...

Let `A ,B ,C` be three events. If the probability of occurring exactly one event out of `Aa n dBi s1-x ,` out of`Ba n dCi s1-2x ,` out of`Ca n dAi s1-x ,` and that of occuring three events simultaneously is `x^2` , then prove that the probability that atleast one out of A, B, C will occur is greaer than 1/2 .

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P(exactly one event out of A and B occurs)
`=P[(AnnB')uu(A'nnB)]`
`=P(A uu B) - P(Ann B)`
= `P(A) + P(B) - 2P(A nn B)`
`therefore P(A) + P(B) - 2P(A nn B) = 1 - a (1)`
Similarly,
`P(B) + P(C ) - 2P(B nn C) = 1 - 2a (2)`
`P(C ) + P(A) - 2P(C nn A) = 1 - a (3)`
`P(A nn B nn C) = a^(2) (4)`
Now,
`P(A uu B uu C) = P(A) + P(B) + P(C) - P(A nn B) - P(B nn C) - P(C nn A) + P(A nn B nn C)`
`=(1)/(2) [P(A) + P(B) - 2P(B nn C) + P(B) + P(C) - 2P(B nn C) + P(C) + P(A) - 2P(C nn A)] + P(A nn B nn C)`
`=(1)/(2) [ 1 - a + 1 - 2a + 1 - a] + a^(2)` [Using Eqs. (1), (2), (3), and (4)]
`=(3)/(2) - 2a + a^(2)`
`=(1)/(2) + (a-1)^(2) gt (1)/(2) " "(because a ne 1)`
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