If A and B are two independent events such that P(A’ ∩ B) = 2/15 and P(A ∩ B’) = 1/6 then find P(A) and P(B).

Basic Idea: If A and B are two independent events then

P(A ∩ B) = P(A)P(B) and A’ and B or B’ and A are also independent and same logic can be applied to them.

Let P(A) = x and P(B) = y

∴ P(A’) = 1 – x and P(B’) = 1 – y

Given that,

A and B are independent. So,

P(A’ ∩ B) = P(A’)P(B) = 2/15

⇒ (1-x)y = 2/15 …(1)

Similarly, P(A ∩ B’) = P(A)P(B’) = 1/6

⇒ x(1-y) = 1/6 …(2)

From equation 1 and 2 we can write as-

⇒

⇒

⇒

⇒

⇒ 180x² + 30 = 186x

⇒ 180x² – 186x + 30 = 0

∴

∴

also x =

∴ P(A) =

Putting x = in equation 1,we get

y =

Hence,

P(A) = 1/5, 5/6

Or P(B) = 1/6, 4/5