Question

The probability of the simultaneous occurrence of at least one of the two events X and Y is a. If the probability that exactly one of the events X and Y occurs is b, prove that P(X′) + P(Y′) = 2 − 2a + b.

Answer 1

P(X ∪ Y) = a

Exactly one event occurs:

P(X ∩ Y′) + P(X′ ∩ Y) = b

P(X′) + P(Y′) = 2 − [P(X) + P(Y)]

P(X ∪ Y) = P(X) + P(Y) − P(X ∩ Y)

b = P(X) + P(Y) − 2P(X ∩ Y)

P(X) + P(Y) = 2a − b

P(X′) + P(Y′) = 2 − (2a − b)

P(X′) + P(Y′) = 2 − (2a + b)

Hence proved.