Question

If A and B are two events such that P(A ∪ B) = 0.7, P(A ∩ B) = 0.2, and P(B) = 0.5 then show that A and B are independent.

Answer 1

Given P(A ∪ B) = 0.7, P(A ∩ B)= 0.2 and P(B) = 0.5 

To find P(A) 

Now, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

(i.e.,) 0.7 = P(A) + 0.5 – 0.2 

⇒ 0.7 – 0.5 + 0.2 = P(A) 

(i.e.,) P(A) = 0.4 

Now P(A ∩ B) = 0.2 … (i) 

P(A) P(B) = 0.4 × 0.5 = 0.2 … (ii) 

(1) = (2) ⇒ P(A ∩ B) = P(A) P(B) 

⇒ A and B are independent.