The probability of event `A` is `3//4`. The probability of event `B`, given that event `A` occurs is `1//4`. The probability of event `A`, given that event `B` occurs is `2//3`. The probability that neither event occurs is

Options

A. `(1)/(6)`
B. `(27)/(112)`
C. `(5)/(32)`
D. `(1)/(8)`

Answer

C

Explanation

`P(A) = (3)/(4) , P(B//4) = (1)/(4) , P(A//B) = (2)/(3) , P(A^(C ) cap B^(C ) ) = ?`
`:. P (B//A) = (P(B cap A))/(P(A)) = (1)/(4)` (given)
` P (B cap A) = (3)/(4) . (1)/(4) =(3)/(16)`
Now , `P(A//B) = ( P(A cap B))/(P(B)) =(2)/(3)` (given )
`:. P(B) =(3)/(2) . (3)/(16) = (9)/(32)`
`:. P (A or B) = (3)/(4) + (9)/(32) -(3)/(16) = (24 + 9-6)/(32) = (27)/(32)`
`:. P(A^(C ) cap B^(C )) = 1- P (A or B) = 1 - (27)/(32) = (5)/(32)`