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

We have, P(A ⋃ B) = P(A ⋂ B)

By General Addition Rule,

P(A) + P (B) – P(A ⋂ B) = P(A ⋃ B)

⇒ P(A) + P (B) – P(A ⋂ B) = P(A ⋂ B) [given]

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

But P(A) – P(A ⋂ B) ≥ 0

and P(B) – P(A ⋂ B) ≥ 0

[∵ P(A ⋂ B) ≤ P(A) or P(B)]

⇒ P(A) – P(A ⋂ B) = 0

and P(B) – P(A ⋂ B) = 0

⇒ P(A) = P(A ⋂ B) …(i)

and P(B) = P(A ⋂ B) …(ii)

From (i) and (ii), we get

∴ P(A) = P(B)

Hence, the correct option is (A).