The Problem of the Points. Two players engage in a game. They are equally matched, and the chances of winning a point are the same for each

Question

The Problem of the Points. Two players engage in a game. They are equally matched, and the chances of winning a point are the same for each player. The first player to score 10 points wins the $100 stakes. The game is interrupted suddenly when Player A has 7 points and Player B has 5 points. How should the stakes be divided fairly

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Nguyệt Ánh 3 years 2021-07-24T04:48:28+00:00 1 Answers 9 views 0

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  1. thienthi
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    2021-07-24T04:49:34+00:00
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    Answer:

    Player A receives $77.34375 of the sakes

    Player B receives $22.65625 of the stakes

    Step-by-step explanation:

    Given that at the time the game was interrupted, we have;

    The number of points player A  has = 7 points

    The number of points player B has = 5 points

    The amount the first player to score 10 points wins = $100

    The number of points remaining for player A to win the game = 3

    The number of points remaining for player B to win the game = 5

    Therefore, the number of trials remaining for a winner to emerge = 3 + 5 – 1 = 7 trials

    We take the probability that Player A wins a point as success

    We find the likelihood of Plater A winning by using binomial theory as follows;

    P(S) = \sum\limits_{j=3}^7 \dbinom{7}{3} \cdot \dfrac{1}{2^7} = \dfrac{35}{128} + \dfrac{35}{128} + \dfrac{21}{128} + \dfrac{7}{128} + \dfrac{1}{128} = \dfrac{99}{128}

    Therefore, given that the likelihood of Player A winning = 99/128

    The stakes should be divided such that Player A gets 99/128 share of the stakes while Player B gets 1 – 99/128 = 29/128 share of the stakes

    Therefore, Player A gets (99/128) × $100 = $77.34375

    Player B gets (29/128) × $100 =  $22.65625

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