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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 11

  • question_answer
    If A and B are events such that \[P(\bar{A}\cup \bar{B})=\frac{3}{4},\] \[P(\bar{A}\cap \bar{B})=\frac{1}{4}\]  and \[P(A)=\frac{1}{3},\] then \[P(\bar{A}\cap B)\] is

    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, \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] \[\Rightarrow \,\,\frac{3}{4}=\frac{1}{3}+P(B)-\frac{1}{4}\] \[\therefore \,\,\,P(B)=\frac{3}{4}-\frac{1}{12}=\frac{2}{3}\] \[\therefore \,\,\,\,\,P(\bar{A}\cap B)=\frac{2}{3}-\frac{1}{4}=\frac{5}{12}\]


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