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BCECE Engineering BCECE Engineering Solved Paper-2010

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

    A)  dependent

    B)  independent

    C)  mutually exclusive

    D)  None of the above

    Correct Answer: B

    Solution :

    We have \[P(A\cup B)=\frac{5}{6},(A\cap B)=\frac{1}{3},P(B)=\frac{1}{2}\] Now,     \[P(B)=1-P(\bar{B})\]                 \[=1-\frac{1}{2}=\frac{1}{2}\] Also by addition theorem,                 \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]                 \[\Rightarrow \]\[P(A)=P(A\cup B)+P(A\cap B)-P(B)\]                 \[\Rightarrow \]\[P(A)=\frac{5}{6}+\frac{1}{3}-\frac{1}{2}\]                 \[\Rightarrow \]\[P(A)=\frac{2}{3}\]                 \[\therefore \]  \[P(A)P(B)=\frac{2}{3}.\frac{1}{2}=\frac{1}{3}\] Thus, \[P(A\cap B)=P(A)P(B)\] Hence, A and B are independent events.


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