Question

A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of the occurrence of A. 

Answer 1

Given that A and B are independent events 

∴ P(A ∩B) = P (A). P(B) . . . . . . . . . . . . . . . . . . . . . . . . . (1) 

Also given that P (A ∩ B) = 1/6 . . . . . . . . . . . . . . . . . . . . . . . . . (2) 

And P(bar A ∩ B) = 1/3  ................(3)

Also P(bar A ∩ B) = 1 - P(A ∪ B)

⇒ P(A ∩ B) = 1 - P(A) - P(B)P(A ∩ B)

⇒ 1/3 = 1 - P(A) - P(B) + 1/6

⇒ P(A) + P(B) ........(4)

From (1) and (2) we get

Let P(A) = x and P(B) = y then eq's (4) (5) become

x + y = 5/6, xy = 1/6

⇒ x - y = (x + y)² - 4xy

= 25/36 - 4/6 = 1/6

∴ We get x = 1/2 and y = 1/3

Or x = 1/3 and y = 1/2

Thus  P(A) = 1/2 and P(B) = 1/3 or P(A) = 1/3 and P(B) = 1/2