A) \[\frac{5}{12}\]
B) \[\frac{3}{8}\]
C) \[\frac{5}{8}\]
D) \[\frac{1}{4}\]
Correct Answer: A
Solution :
[a] \[P(\bar{A}\cup \bar{B})=\frac{3}{4}\Rightarrow P(\overline{A\cap B})=\frac{3}{4}\Rightarrow P(A\cap B)=\frac{1}{4}\] \[P(\bar{A}\cap \bar{B})=\frac{1}{4}\Rightarrow P(\overline{A\cup B})=\frac{1}{4}\Rightarrow P(A\cup B)=\frac{3}{4}\] \[P(A)=\frac{1}{3}\,\,\,\,\therefore \,\,\,P(\bar{A})=\frac{2}{3}\] Now, \[P(\bar{A}\cap B)=P(B)-P(A\cap B)\] Also, \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] \[\Rightarrow \,\,\frac{3}{4}=\frac{1}{3}+P(B)-\frac{1}{4}\] \[\therefore \,\,\,P(B)=\frac{3}{4}-\frac{1}{12}=\frac{2}{3}\] \[\therefore \,\,\,\,\,P(\bar{A}\cap B)=\frac{2}{3}-\frac{1}{4}=\frac{5}{12}\]You need to login to perform this action.
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