A) \[\frac{1}{2},\frac{3}{2}\]
B) \[\frac{1}{2},\frac{1}{3}\]
C) not possible
D) None of these
Correct Answer: B
Solution :
Given that, \[P(A\cap B)=\frac{1}{6}\] \[\Rightarrow \] \[P(A)\,\,P(B)=\frac{1}{6}\] ?.. (i) and \[P(\bar{A}\cap \bar{B})=\frac{1}{3}\] \[\Rightarrow \] \[P(\bar{A})P(\bar{B})=\frac{1}{3}\] \[\Rightarrow \] \[(1-P(A))\,\,(1-P(B))=\frac{1}{3}\] \[\Rightarrow \] \[1-\frac{1}{3}+P(A)\,P(B)=P(A)+P(B)\] \[\Rightarrow \] \[\frac{2}{3}+\frac{1}{6}=P(A)+P(B)\] \[\Rightarrow \] \[P(A)+P(B)=\frac{5}{6}\] On solving Eqs. (i) and (ii), we get \[P(A)=\frac{1}{2},\,\,P(B)=\frac{1}{3}\] or \[P(A)=\frac{1}{3},\,\,P(B)=\frac{1}{2}\]
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