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JEE Main & Advanced Mathematics Question Bank Critical Thinking

  • question_answer
    If A and B are two events such that \[P\,(A\cup B)=P\,(A\cap B),\] then the true relation is                                          [IIT 1985]

    A)                 \[P\,(A)+P\,(B)=0\]              

    B)                 \[P\,(A)+P\,(B)=P\,(A)\,P\,\left( \frac{B}{A} \right)\]

    C)                 \[P\,(A)+P\,(B)=2\,P\,(A)\,P\,\left( \frac{B}{A} \right)\]

    D)                 None of these

    Correct Answer: C

    Solution :

               \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]            \[\Rightarrow P(A\cap B)=P(A)+P(B)-P(A\cap B)\]            \[\{\because \,\,\,P(A\cap B)=P(A\cup B)\}\]            \[\Rightarrow 2P(A\cap B)=P(A)+P(B)\]            \[\Rightarrow 2P(A)\,.\,\frac{P(A\cap B)}{P(A)}=P(A)+P(B)\]                 \[\Rightarrow 2P(A).P\left( \frac{B}{A} \right)=P(A)+P(B)\].


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