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JEE Main & Advanced Mathematics Probability Question Bank Odds in favour and odds against Addition theorem on probability

  • question_answer
    If A and B are events such that \[P(A\cup B)=3/4,\] \[P(A\cap B)=1/4,\] \[P(\bar{A})=2/3,\] then \[P(\bar{A}\cap B)\] is [AIEEE 2002]

    A)                 \[\frac{5}{12}\]    

    B)                 \[\frac{3}{8}\]

    C)                 \[\frac{5}{8}\]       

    D)                 \[\frac{1}{4}\]

    Correct Answer: A

    Solution :

                       \[P\,(A\cup B)\,=\frac{3}{4},\] \[P(A\cap B)=\frac{1}{4}\]                    \[P(\bar{A})\,=\frac{2}{3}\,\,\Rightarrow \] \[P(A)=\,\frac{1}{3}\]                    \[\therefore \]\[P(A\cap B)\,=\,P(A)\,+P\,(B)\,-P(A\cup B)\]                    \[\frac{1}{4}=\frac{1}{3}+P(B)\,-\frac{3}{4}\] Þ \[P(B)\,=\frac{2}{3}\]                                 \[P(\bar{A}\cap \,B)\]= \[P(B)-\,P\,(A\cap B)\]\[=\frac{2}{3}-\frac{1}{4}=\frac{8-3}{12}=\frac{5}{12}.\]


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