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

  • question_answer
    The probability that a man will be alive in 20 years is \[\frac{3}{5}\] and the probability that his wife will be alive in 20 years is \[\frac{2}{3}\]. Then the probability that at least one will be alive in 20 years, is                                   [Bihar CEE 1994]

    A)                 \[\frac{13}{15}\]  

    B)                 \[\frac{7}{15}\]

    C)                 \[\frac{4}{15}\]    

    D)                 None of these

    Correct Answer: A

    Solution :

                       Let \[A\] be the event that the husband will be alive 20 years. \[B\] be the event that the wife will be alive 20 years. Clearly \[A\] and \[B\] are independent events                    \ \[P(A\cap B)=P(A).\,P(B)\]                    Given \[P(A)=\frac{3}{5},\] \[P(B)=\frac{2}{3}\]            The probability that at least of them will be alive 20 years is \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]                    \[=P(A)+P(B)-P(A)P(B)=\frac{3}{5}+\frac{2}{3}-\frac{3}{5}.\frac{2}{3}=\frac{9+10-6}{15}=\frac{13}{15}\]                 Aliter : Required probability is \[1-P(A\]and \[B\] both will die) \[=1-\frac{2}{5}\times \frac{1}{3}=1-\frac{2}{15}=\frac{13}{15}.\]      


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