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

  • question_answer
    If E and F are independent events such that \[0<P(E)<1\] and \[0<P\,(F)<1,\] then                              [IIT 1989]

    A)                 E and \[{{F}^{c}}\](the complement of the event F) are independent

    B)                 \[{{E}^{c}}\]and \[{{F}^{c}}\]are independent

    C)                 \[P\,\left( \frac{E}{F} \right)+P\,\left( \frac{{{E}^{c}}}{{{F}^{c}}} \right)=1\]

    D)                 All of the above

    Correct Answer: D

    Solution :

                      \[P(E\cap F)=P(E)\,.\,P(F)\]            Now,\[P(E\cap {{F}^{c}})=P(E)-P(E\cap F)=P(E)[1-P(F)]=P(E)\,.P({{F}^{c}})\]            and \[P({{E}^{c}}\cap {{F}^{c}})=1-P(E\cup F)=1-[P(E)+P(F)-P(E\cap F)\]                              \[=[1-P(E)][1-P(F)]=P({{E}^{c}})\,P({{F}^{c}})\]            Also \[P(E/F)=P(E)\] and \[P({{E}^{c}}/{{F}^{c}})=P({{E}^{c}})\]                 \[\Rightarrow P(E/F)+P({{E}^{c}}/{{F}^{c}})=1.\]


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