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JEE Main & Advanced Mathematics Question Bank Critical Thinking

  • question_answer
    A bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag replacing the ball each time after the draw till one of them draws a white ball and wins the game. A begins the game. If the probability of A winning the game is three times that of B, then the ratio a : b is

    A)                 1 : 1       

    B)                 1 : 2

    C)                 2 : 1       

    D)                 None of these

    Correct Answer: C

    Solution :

               Let \[W\] denote the event of drawing a white ball at any draw and \[B\] that for a black ball.            Then \[P(W)=\frac{a}{a+b},\] \[P(B)=\frac{b}{a+b}\]            \[P(A\] wins the game)\[=P(W\]or\[BBW\]or\[BBBBW\]or ?..)            \[=P(W)+P(B)P(B)P(W)+P(B)P(B)P(B)P(B)P(W)+\,......\]            \[=\frac{P(W)}{1-P{{(B)}^{2}}}=\frac{\frac{a}{a+b}}{1-\frac{{{b}^{2}}}{{{(a+b)}^{2}}}}=\frac{a(a+b)}{{{a}^{2}}+2ab}=\frac{(a+b)}{a+2b}\]            Also \[P(B\] wins the game)\[=1-\frac{a+b}{a+2b}=\frac{b}{a+2b}\]            According to the given condition,                                 \[\frac{a+b}{a+2b}=3.\frac{b}{a+2b}\Rightarrow a=2b\Rightarrow a:b=2:1.\]


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