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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

    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 the above

    Correct Answer: D

    Solution :

                      \[A,\,\,B\] are independent as \[P(A)=P\left( \frac{A}{B} \right)\]            \[P\left( \frac{{{A}'}}{B} \right)=1-\frac{1}{4}=\frac{3}{4}\] as \[A,\,\,B\] are independent            \[\Rightarrow {A}',\,B\] are independent.                 \[P\left( \frac{{{B}'}}{{{A}'}} \right)=P({B}')=1-\frac{1}{2}=\frac{1}{2}.\]


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