Question

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′) = ?
1. 2 - 2q + p.
2. p + q + 2.
3. 2 - 2p + q.
4. 2(p + q).

Answer 1

Correct Answer - Option 3 : 2 - 2p + q.

Concept:

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• P(A') = 1 - P(A)

 

Calculation:

Given: Probability of simultaneous occurrence of at least one of two events A and B is p

So, P(A ∪ B) = p.

Since probability that exactly one of A, B occurs = q (given), we get:
P(A ∪ B) - P(A ∩ B) = q
⇒ p - P(A ∩ B) = q
⇒ P(A ∩ B) = p - q

As we know, 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

Now, P (A′) + P (B′) = 1 - P(A) + 1 - P(B)

= 2 - [P(A) + P(B)]

= 2 - (2p - q)

= 2 - 2p + q.