The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then P (A′) + P (B′) =?

By BYJU'S Exam Prep

Updated on: September 25th, 2023

2 – 2q + p
p + q + 2
2 – 2p + q
2(p + q)

The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then P (A′) + P (B′) = 2 – 2p + q.

P(A’) = 1 – P(A)

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

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1. The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then P (A′) + P (B′) =?

It is given that

The probability of simultaneous occurrence of at least one of two events A and B is p

P(A ∪ B) = p

Due to the probability that just one of A, B occurs = q

So we get

P(A ∪ B) – P(A ∩ B) = q

p – P(A ∩ B) = q

P(A ∩ B) = p – q

We know that,

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

p = P(A) + P(B) – (p – q)

P(A) + P(B) = p + (p – q)

P(A) + P(B) = 2p – q

Similarly

P (A′) + P (B′) = 1 – P(A) + 1 – P(B)

= 2 – [P(A) + P(B)]

= 2 – (2p – q)

= 2 – 2p + q.

Summary: