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JEE Main & Advanced Mathematics Probability Question Bank Conditional probability bayes theorem

  • question_answer
    For two events A and B, if \[P(A)=P\left( \frac{A}{B} \right)=\frac{1}{4}\] and \[P\left( \frac{B}{A} \right)=\frac{1}{2}\], then                                          [MP PET 2003]

    A)                 A and B are independent    

    B)                 \[P\left( \frac{{{A}'}}{B} \right)=\frac{3}{4}\]

    C)                 \[P\left( \frac{{{B}'}}{{{A}'}} \right)=\frac{1}{2}\]        

    D)                 All of these

    Correct Answer: D

    Solution :

               \[P\left( \frac{B}{A} \right)=\frac{1}{2}\]\[\Rightarrow \,\frac{P(B\cap A)}{P(A)}=\frac{1}{2}\]\[\Rightarrow P(B\cap A)=\frac{1}{8}\]            \[P\left( \frac{A}{B} \right)=\frac{1}{4}\]\[\Rightarrow \,\frac{P(A\cap B)}{P(B)}=\frac{1}{4}\] \[\Rightarrow P(B)\,=\frac{1}{2}\]            \[P(A\cap B)=\frac{1}{8}\,=P(A).\,P(B)\,\]            \[\therefore \] Events A and B are independent.            Now, \[P\,\left( \frac{{{A}'}}{B} \right)=\frac{P({A}'\cap B)}{P(B)}=\frac{P({A}')\,P(B)}{P(B)}=\frac{3}{4}\]                 and \[P\,\left( \frac{B'}{A'} \right)=\frac{P(B'\cap A')}{P(A')}=\frac{P(B')\,P(A')}{P(A')}=\frac{1}{2}\].


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