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JCECE Engineering JCECE Engineering Solved Paper-2010

  • question_answer
    If \[A\] and \[B\] are two independent events such that \[P(A)=\frac{1}{2}\] and \[P(B)=\frac{1}{5},\] then which of the following is/are correct?

    A) \[P(A\cup B)=\frac{5}{3}\]          

    B) \[P\left( \frac{A}{B} \right)=\frac{1}{3}\]

    C) \[P\left( \frac{A}{A\cup B} \right)=\frac{5}{6}\]

    D)  All of these

    Correct Answer: C

    Solution :

    Since, \[A\] and \[B\] are independent events \[\therefore \]  \[P(A\cap B)=P(A)P(B)\]                                    \[=\frac{1}{2}\cdot \frac{1}{5}=\frac{1}{10}\] Now,\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\] \[\Rightarrow \]               \[P(A\cup B)=\frac{1}{2}+\frac{1}{5}-\frac{1}{10}\] \[\Rightarrow \]               \[P(A\cup B)=\frac{5+2-1}{10}=\frac{6}{10}=\frac{3}{5}\] and        \[P\left( \frac{A}{B} \right)=P(A)\]                 (since, \[A\] and \[B\] are independent) \[\Rightarrow \]               \[P\left( \frac{A}{B} \right)=\frac{1}{2}\]               \[\left[ \because \,\,P(A)=\frac{1}{2} \right]\] and\[P\left( \frac{A}{A\cup B} \right)=\frac{P[A\cap (A\cup B)]}{P(A\cup B)}=\frac{P(A)}{P(A\cup B)}\]                 \[P\left( \frac{A}{A\cup B} \right)=\frac{\frac{1}{2}}{\frac{3}{5}}=\frac{5}{6}\]


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