For two events A and B let, `P(A)=(3)/(5), P(B)=(2)/(3)`, then which of the following is/are correct ?

Options

A. `P(A cup B) ge 2//3`
B. `4//15 le P(A cap B) le 3//5`
C. `2//5 le P(A//B) le 9//10`
D. `P(A cap overline(B)) ge 1//3`

Answer

D

Explanation

Since `B sube A cup B`. Therefore,
`P(B) le P(A cup B) implies P(A cup B) gt 2//3`.
So, alternative (a) is correct.
Now,
`P(A cap B) =P(A)+P(B)-P(A cup B)`
`implies P(A cap B) ge P(A)+P(B)-1`
`implies P(A cap B) ge (3)/(5)+(2)/(3)-1=(4)/(15)`
Also,
`A cap B sube A`
`implies P(A cap B) le P(A) implies (A cap B) le 3//5`
`therefore (4)/(15) le P(A cap B) le (3)/(5)`
So, alternative (b) is correct.
Now,
`(4)/(15) le P(A cap B) le (3)/(5)`
`implies (4)/(15) le P(A//B)P(B) le (3)/(5)`
`implies (4)/(15) le (2)/(3) P(A//B) le (3)/(5) implies (2)/(5) le P(A//B) le (9)/(10)`
So, alternative (c ) is correct.
Finally,
`P(A cap overline(B))=P(A)-P(A cap B) le (3)/(5)-(4)/(15)=(1)/(3)`
So, alternative (d) is also incorrect.