? 發問於 科學及數學數學 · 1 十年前

Expected gain

2 players put one dollar into a pot. They decide to throw a pair of dice alternately. The first one who throws a total of 5 on both dice wins the pot. How much should the player who starts add to the pot to make it a fair game?

Is the answer = 1 / 8 one eighth?

How to think the equation?

2 個解答

評分
  • 1 十年前
    最愛解答

    P(total of 5 in 1 throw) = 4/36 = 1/9 = p

    Let player A starts first.

    P(A wins the pot in 1st try) = p = 1/9

    P(B wins the pot in 1st try) = (1-p)p = 8/81

    P(nobody wins the pot in 1st try) = 1-p-(1-p)p = (1-p)^2 = 64/81

    P(A wins the pot) = 1/9 + 64/81 * 1/9 + (64/81)^2 * 1/9 + ... = 1/9 [1 + 64/81 + (64*81)^2+...]

    P(B wins the pot) = 8/81 + 64*81 * 8/81 + (64/81)^2 * 8/81 + ... = 8/81 [1 + 64/81 + (64/81)^2 + ... ]

    Let x = [1 + 64/81 + (64/81)^2 + ... ]

    P(A wins the pot) + P(B wins the pot) = 1

    1/9 x + 8/81 x = 1

    9/81 x + 8/81 x = 1

    17/81 x = 1

    x = 81/17

    P(A wins the pot) = 9/17

    P(B wins the pot) = 8/17

    Let $y be the first player's extra amount of the bet.

    Expect gain of A = Expect gain of B = zero

    9/17 * 1 - 8/17 * (1 + y) = 8/17 * (1 + y) - 9/17 * 1 = 0

    8 * (1 + y) - 9 = 0

    1 + y = 9/8

    y = 1/8

    Therefore, player A needs to put an extra 1/8 dollar to make the game fair.

    資料來源: my mathematical knowledge
  • Ivan
    Lv 5
    1 十年前

    haha I saw you mention my name in some post 2 months ago, then answer you la :P

    Ivan and Jessica is playing, Ivan play first. Ivan add $x to the pot. so if Ivan win he will gain $1 and if Jessica win she will gain $1+x.

    Probability to throw 5 is 4/36 = 1/9

    Probability for Ivan to win is: (first time 5) + (not 5, not 5, is 5) + (not 5, not 5, not 5, not 5, is 5) +......... = 1/9 + 8/9*8/9*1/9 + (8/9)^4 * 1/9 +........

    = 1/9 Sum (64/81)^n

    = 1/9 * (1/ (1- 64/81)) = 9/17

    So Ivan has probability 9/17 to win, and so Jessica has probability 8/17 to win.

    To be a fair game, the expected gain for both people should be the same:

    9/17 (1) = 8/17(1+x)

    so solve for x, Ivan need to add $1/8 more.

    資料來源: I wont play this game though :P
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