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JEE Main & Advanced Mathematics Question Bank Critical Thinking

  • question_answer
    If A and B are two events such that \[P(A)=\frac{1}{2}\]and \[P(B)=\frac{2}{3},\] then

    A)                 \[P\,(A\cup B)\ge \frac{2}{3}\]            

    B)                 \[\frac{1}{6}\le P(A\cap B)\le \frac{1}{2}\]

    C)                 \[\frac{1}{6}\le P({A}'\cap B)\le \frac{1}{2}\]  

    D)                 All of the above

    Correct Answer: D

    Solution :

               We have \[P(A\cup B)\ge \max .\]\[\{P(A),\,P(B)=\frac{2}{3}\]            \[P(A\cap B)\le \min .\]\[\{P(A),P(B)\}=\frac{1}{2}\]            and\[P(A\cap B)=P(A)+P(B)-P(A\cup B)\ge P(A)-P(B)-1=\frac{1}{6}\]            \[\Rightarrow \frac{1}{6}\le P(A\cap B)\le \frac{1}{2}\]            \[P\,({A}'\cap B)=P(B)-P(A\cap B)\]            Hence \[\frac{2}{3}-\frac{1}{2}\le P({A}'\cap B)\le \frac{2}{3}-\frac{1}{6}\]                 \[\Rightarrow \frac{1}{6}\le P({A}'\cap B)\le \frac{1}{2}\].


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