Question

If P(A ∪ B) = P(A ∩ B) for any two events A and B, then ______.

Options

• P(A) = P(B)

• P(A) > P(B)

• P(A) < P(B)

• None of these

Answer 1

If P(A ∪ B) = P(A ∩ B) for any two events A and B, then P(A) = P(B).

Explanation:

Given that: P(A ∪ B) = P(A ∩ B)

⇒ P(A) + P(B) – P(A ∩ B) = P(A ∩ B)

⇒ [P(A) – P(A ∩ B)] + [P(B) – P(A ∩ B)] = 0

But P(A) – P(A ∩ B) ≥ 0     ......(i)   .......[∵ P(A ∩ B) ≤ P(A) or P(B)]

And P(B) – P(A ∩ B) ≥ 0   ......(ii)

From equation (i) and (ii) we get

P(A) = P(B)