If `P(B)=(3)/(4), P(AcapBoverline(C))=(1)/(3) and P(overline(A)capoverline(B)capoverline(C))=(1)/(3)` , then `P(AcapC)` is equal to

Options

A. `(1)/(2)`
B. `(1)/(6)`
C. `(1)/(15)`
D. `(1)/(9)`

Answer

A

Explanation

We have,
`P(overline(A) cap B cap overline(C ))=(1)/(3)`
`implies P((B cap overline(C )) cap overline(A))=(1)/(3)`
`implies P(B cap overline(C ))-P((B cap overline(C )) cap A)=(1)/(3) " " [because P(X cap overline(Y))=P(X)-P(X cap Y)]`
`implies P(B cap overline(C ))-P(A cap B cap overline(C ))=(1)/(3)`
`implies P(B cap overline(C ))-(1)/(3)=(1)/(3)`
` implies P(B cap overline(C ))=(2)/(3)`
`implies P(B)-P(B cap C)=(2)/(3)`
`implies P(B cap C)=P(B)-(2)/(3)=(3)/(4)-(2)/(3)=(1)/(12)`