For the three events A,B and C,P(exactly one of the events A or B occurs) = P (exactly one of the events C or A occurs) = P(exactly one of the events B or C occurs) = P and P( all three events occur simultaneously) = P2, where Then find the probability of atleast one of the three events A,B and C occurring.

Engineering

Mathematics

Conditional Probability

Question

For the three events A,B and C,P(exactly one of the events A or B occurs) = P (exactly one of the events C or A occurs) = P(exactly one of the events B or C occurs) = P and P( all three events occur simultaneously) = P², where $0 < p < \frac{1}{2}$ Then find the probability of atleast one of the three events A,B and C occurring.

$\frac{3 p + 2 p^{2}}{2}$

$\frac{3 p - 2 p^{2}}{2}$

$\frac{2 p + 3 p^{2}}{2}$

$\frac{2 p - 3 p^{2}}{2}$

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According to the question,
$P(A) + P(B) - 2P(A⋂B) = P(B) + P(C) - 2P(B⋂C) = P(A) + P(C) - 2P(A⋂C) = P$ .
Adding, we get, 2(P(A) + P(B) + P(C) – P(A⋂B), – P(B⋂C) – P(A⋂C)) = 3p

$P (A) + P (B) + P (C) - P (A \cap B) - P (B \cap C) - P (A \cap C) = \frac{3 p}{2}$ .

Also P(A⋂B⋂C) = p²
∴ P(AUBUC) = P(A) + P(B) + P(C) – P(A⋂B), – P(B⋂C) – P(A⋂C), + P(AUBUC)

= \frac{3 p}{2} + p^{2} = \frac{3 p + 2 p^{2}}{2}

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