Question

For three events A, B, C

P(exactly one of A or B occur) = P 

P(exactly one of B or C occur) = P

P(exactly one of C or A occur) = P

And P(all three occur) = P². Where 0 < P < (1/2). Then probability of at least one of the three occur is ______

(a) [(3P + 2P²)/2] 

(b) [(P + 3P²)/4]

(c) [(P + 3P²)/2]

(d) [(3P + 2P²)/4]

Answer 1

The correct option (a) [(3P + 2P²)/2] 

Explanation:

 given: P(exactly one of A or B) = P

i.e. P(A) + P(B) – 2P(A ∩ B) = P ....(1)

similarly P(B) + P(C) – 2P(B ∩ C) = P .....(2)

P(C) + P(A) – 2P(C ∩ A) = P .....(3)

(1) + (2) + (3) gives, 

2[P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(C ∩ A)] = 3P

∴ P(A ∪ B ∪ C) – P(A ∩ B ∩ C) = (3P/2)

∴ P(A ∪ B ∪ C) – P² = (3P/2) -------- given: P(A ∩ B ∩ C) = P²

∴ P(A ∪ B ∪ C) = (3P/2) + P² = [(3P + 2P²)/2].