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. equally likely but not independent equally likely and mutually exclusive
B. equually like and mutually exclusive
C. Mutually exclusive and independent
D. independent but not equally likely

Answer

D

Explanation

We have, `P(bar(AuuB))=1/4,P(AnnB)=1/4`
`P(barA)=1/4impliesP(A)=1-1/4=3/4`
`thereforeP(bar(AuuB))=1-P(auuB)=1-[P(A)+P(B)-P(AnnB)]`
`impliesimplies1/6=1-3/4-P(B)+1/4`
`impliesP(B)=1-1/2-1/6=(6-3-1)/(6)=2/6=1/3`
Since `P(AnnB)=P(A)P(B)and P(A)neP(B),` therefore A and B are independent but not equally likely.