If A and B are events such that `P(A uu B) = (3)//(4), P(A nn B) = (1)//(4)` and `P(A^(c)) = (2)//(3)`, then find
(a) P(A) (b) P(B)
(c ) `P(A nn B^(c )) (d) P(A^(c ) nn B)`

Answer

`P(A) = 1//3; P(B) = 2//3; P(A nn B^(c )) = 1//12; P(A^(c )nn B) = 5//12`

Explanation

`P(A^(c)) = 1 - P(A) or P(A) = 1 - P(A^(c)) = 1 - 2//3 = 1//3`
`P(A uu B) = P(A) + P(B) - P(A nn B)`
`implies 3//4 = 1//3 + P(B) - 1//4 implies P(B) = 2//3`
`P(A nn B^(c)) = P(A) - P(A nn B) = (1)/(3) - (1)/(4) = (1)/(12)`
`P(A^(c) nn B) = P(B) - P(A nn B) = (2)/(3) - (1)/(4) = (5)/(12)`