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匿名
匿名 發問於 科學及數學數學 · 10 月前

expected value(find minimum entrance fee of the game)*急需?

A player pays an entrance fee of $k to join a game. The player selects one number from 1, 2, 3 or 4 and rolls three fair 4-sided dice.

If the chosen number appears on all three dice, the player wins $4k.If the number appears on exactly two of the dice, the player wins $3k.If the number appears on exactly one die, the player wins $30.If the number does not appear on any of the dice, the player wins nothing.The game organiser wants to make a positive profit for the game. Find the minimum entrance fee of the game. 

1 個解答

評分
  • 匿名
    10 月前
    最愛解答

    Question:

    A player pays an entrance fee of $k to join a game. The player selects one number from 1, 2, 3 or 4 and rolls three fair 4-sided dice.

    If the chosen number appears on all three dice, the player wins $4k.

    If the number appears on exactly two of the dice, the player wins $3k.

    If the number appears on exactly one die, the player wins $30.

    If the number does not appear on any of the dice, the player wins nothing.

    The game organiser wants to make a positive profit for the game. Find the minimum entrance fee of the game. 

    🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘

    Assumption:

    Each die has these four numbers 1, 2, 3 and 4.

    🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘🔘

    Solution:

    The probability that the player wins $4k

    = (1/4) (1/4)³ + (1/4) (1/4)³ + (1/4) (1/4)³ + (1/4) (1/4)³

    = 4 (1/4) (1/4)³

    = 1/64

    The probability that the player wins $3k

    = 4 (1/4) [ (1/4) (1/4) (1 - 1/4) + (1/4) (1 - 1/4) (1/4) + (1 - 1/4) (1/4) (1/4) ]

    = 9/64

    The probability that the player wins $30

    = 4 (1/4) [ (1/4) (1 - 1/4) (1 - 1/4) + (1 - 1/4) (1/4) (1 - 1/4) + (1 - 1/4) (1 - 1/4) (1/4) ]

    = 27/64

     

    The probability that the player wins nothing

    = 4 (1/4) (3/4)³

    = 27/64

    If the game organiser wants to make a positive profit for the game,

    ($4k) (1/64) + ($3k) (9/64) + ($30) (27/64) + ($0) (27/64) < $k

    (31k + 810)/64 < k

    k > 270/11 ...... ( ≈ 24.545454 )

    ∴ The minimum entrance fee of the game = $25.

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