A and B are two independent events such that `0 lt P(A) lt 1,0 < P(B) lt 1 ` then which of the following is not correct ?

Options

A. A and B are mutually exclusive
B. A and `overline(B)` are independent
C. `overline(A) " and " overline (B)` are independent
D. `P(A//B)+P(overline(A)//B)=1`.

Answer

A

Explanation

Since A and B are independent events.
`therefore P(A cap B)=P(A)P(B) ne 0`
So, A and B cannot be mutually exclusive events.
Now,
`P(A cap overline(B))=P(A)-P(A cap B)`
`implies P(A cap overline(B))=P(A)-P(A)P(B)`
`implies P(A cap overline(B))=P(A){1-P(B)}=P(A)P(overline(B))`
and,
`P(overline(A) cap overline(B))=1-P(A cup B)`
`implies P(overline(A) cap overline(B))=1-{1-P(overline(A))P(overline(B))} [because` A and B are independent]
`implies P(overline(A) cap overline(B))=P(overline(A))P(overline(B))`
So, A and `overline(B)` as well as `overline(A) " and " overline (B)` are independent events.
Finally,
`P(A//B)+P(overline(A)//B)=(P(A cap B))/(P(B))+(P(overline(A) cap B))/(P(B))`
`implies P(A//B)+P(overline(A)//B)=(P(A cap B)+P(overline(A) cap B))/(P(B))=(P(B))/(P(B))=1`
Hence, alternative (a) is incorrect.