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# 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

• 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.

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Assumption：

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

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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.