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JEE Main & Advanced Mathematics Probability Question Bank Odds in favour and odds against Addition theorem on probability

  • question_answer
    If two events A and B are such that                                                                                   \[P\,(A+B)=\frac{5}{6},\]                                                                                      \[P\,(AB)=\frac{1}{3}\,\] and                                                                                 \[P\,(\bar{A})=\frac{1}{2},\]  then the events A and B are

    A)                 Independent

    B)                 Mutually exclusive

    C)                 Mutually exclusive and independent

    D)                 None of these

    Correct Answer: A

    Solution :

                       We have                                                                               \[P(A+B)=P(A)+P(B)-P(AB)\]                                                                                                         \[\Rightarrow \frac{5}{6}=\frac{1}{2}+P(B)-\frac{1}{3}\Rightarrow P(B)=\frac{4}{6}=\frac{2}{3}\]                    Thus,                                               \[P(A)\,.\,P(B)=\frac{1}{2}\times \frac{2}{3}=\frac{1}{3}=P(AB)\]                                 Hence events                                                                                                     \[A\]  and                                                                                                     \[B\]  are independent.


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