A and B are two independent events. The probabilit

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.

Jitender Pal · Accepted Answer

Hello Student,

Please find the answer to your question

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 ( $\underset{A}{\rightarrow} \ \cap \underset{B}{\rightarrow}$ ) = 1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Also P ( $\underset{A}{\rightarrow} \ \cap \underset{B}{\rightarrow}$ ) = 1 – P ( $\underset{A}{\rightarrow} \ \cup \underset{B}{\rightarrow}$ )

⇒ P ( $\underset{A}{\rightarrow} \ \cap \underset{B}{\rightarrow}$ ) = 1 – P (A) - P (B) P ( $\underset{A}{\rightarrow} \ \cap \underset{B}{\rightarrow}$ )

⇒ 1/3 = 1 - P (A) - P (B) + 1/6

⇒ P (A) + P (B) . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

From (1) and (2) we get

P (A). P (B) = 1/6 . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

Let P (A) = x and P (B) = y then eq’ s (4) and (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 .

Thanks
Jitender Pal
askIITians Faculty

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