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KVPY Sample Paper KVPY Stream-SX Model Paper-13

  • question_answer
    There are 6 boxes labelled \[{{B}_{1}},{{B}_{2}},{{B}_{3}},...,{{B}_{6}}.\] By In each trial two fair dice \[{{D}_{1}},{{D}_{2}}\] are thrown. If \[{{D}_{1}}\] shows j and \[{{D}_{2}}\]shows k, then j balls are put into the box \[{{B}_{K}},\] After n trials, what is the probability that \[{{B}_{1}}\]contains at most one ball?

    A) \[\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)+\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)\left( \frac{1}{6} \right)\]

    B) \[\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)+\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{6} \right)\]

    C) \[\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)+n\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{6} \right)\]

    D) \[\left( \frac{{{5}^{n}}}{{{6}^{n}}} \right)+n\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{{{6}^{2}}} \right)\]

    Correct Answer: D

    Solution :

    \[{{B}_{1}}{{B}_{2}}....{{B}_{n}}\]
    Case I Di never show 1
    Probability \[={{\left( \frac{5}{6} \right)}^{n}}\]
    Case II \[{{D}_{2}}\]shows 1 (one time) then \[{{D}_{1}}\]
    Probability \[=\left\{ {}^{n}{{C}_{1}}{{\left( \frac{5}{6} \right)}^{n-1}}\left( \frac{1}{6} \right) \right\}\left( \frac{1}{6} \right)\]
    Total probability \[={{\left( \frac{5}{6} \right)}^{n}}+n\left( \frac{{{5}^{n-1}}}{{{6}^{n-1}}} \right)\left( \frac{1}{{{6}^{2}}} \right)\]


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