Let A and B be two events such that `p( bar (AuuB))=1/6, p(AnnB)=1/4` and `p( bar A)=1/4` , where ` bar A` stands for the complement of the event A. Then the events A and B are (1) mutually exclusive and independent (2) equally likely but not independent (3) independent but not equally likely (4) independent and equally likely

Options

A. independent but not equally likely
B. independent and equally likely
C. mutually exclusive and independent
D. equally likely but not independent

Answer

A

Explanation

We have,
`P(overline(A cup B))=(1)/(6),P(A cap B)=(1)/(4) " and " P(overline(A))=(1)/(4)`
`implies 1-P(A cup B)=(1)/(6) implies (A cup B) =(5)/(6)`
`implies P(A)+P(B)-P(A cap B) =(5)/(6) implies (3)/(4)+P(B)-(1)/(4)=(5)/(6)`
`implies P(B)=(1)/(3)`
Now, `P(A cap B)=(1)/(4),P(A)=(3)/(4) " and " P(B)=(1)/(3)`
`implies P(A cap B)=P(A)P(B) " and " P(A) ne P(B)`
`implies` A and B independen but not equally likely.