Question

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

(a) P(A) = P(B) (b) P(A) > P(B) (c) P(A) < P(B) (d) None of these

Answer 1

For two events A and B,
P(A∪B) = P(A∩B) (given)
Since A ≤ A∪B
⇒ P(A) ≤ P(A∪B) = P(A∩B)
and A∩B ≤ A
⇒ P(A∩B) ≤ P(A)
⇒ P(A∩B) = P(A)
and
Similarly, A∩B ≤ B ≤ A∪B
⇒ P(A∩B) ≤ P(B) ≤ P(A∪B)
⇒ P(B) = P(A∩B) = P(A∪B)
Hence, P(A) = P(B)
Hence, the correct answer is option A.