`Aa n dB` are two independent events. The probability that both `Aa n dB` occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of the occurrence of `Adot`

Options

A. 0 or 1
B. `(1)/(2) or (1)/(3)`
C. `(1)/(2) or (1)/(4)`
D. `(1)/(3) or (1)/(4)`

Answer

B

Explanation

It is given `P(A nn B) = (1)/(6) and P(A' nn B) = (1)/(3)`
Now, `P(A' nn B') = P(A' nn B') = (1)/(3)`
`rArr 1-P(Auu B)' = P(A' nn B') = (2)/(3)`
But `P(A uu B) = P(A) + P(B) -P(AnnB)`
`rArr P(A) + P(B) = P(A uu B) + P(A nn B)`
`rArr P(A) + P(B) = (5)/(6)`
`because` A and B are independent events.
`P(A nn B) = P(A) P(B) rArr P(A) P(B) = (1)/(6)`
`[P(A) -P(B)]^(2) = [P(A) + P(B)]^(2)-4P(A)P*B)`
` = ((5)/(6))^(2) - 4((1)/(6)) = (25)/(36) - (4)/(6) = (1)/(36)`
`rArr P(A) - P(B) = pm (1)/(6)`
On solving Eqs. (i) and (ii), we get ,brgt `P(A) = (1)/(2) or (1)/(3)`