Two events A and B are mutually not exclusive events if P(A)=1/4,P(B)=2/5 and `P(A cup B)`=1/2 then find `P(A cap B^c)`

Options

A. P(A)≤1-P( B )
B. `P\left(\bar{A}\cap\bar{B}\right)=P\left(\bar{A}\right)-P\left(B\right)`
C. `[-(7)/(9),(4)/(9)]`
D. `[(1)/(3),(2)/(3)]`

Answer

B

Explanation

We must have,
`0 le P(A) le 1`
implies `0 le (3x + 1)/(3) le 1`
implies `(-1)/(3) le x le (2)/(3)" (1)"`
and `0le P(B) le 1`
implies `0 le (1-x)/(4) le 1`
implies `-3 le x le 1 " (2)"`
and `0le P(A uu B) = P(A) + P(B) le 1`
`implies 0 le (3x + 1)/(3) + (1 - x)/(4) le 1`
implies `(-7)/(9) le x le (5)/(9) " (3)"`
From (1), (2) and (3), we get x `in [(-1)/(3), (5)/(9)]`.