皇家卡牌游戏

Royal Card Game

专题: Probability / 概率
难度: L3
来源: OpenQuant

题目详情

我给你一副洗好的 $14$ 卡牌，其中包含每个等级的一张牌和一张小丑牌。如果你翻转了国王、王后和杰克，你将赢得 100 美元，但如果你在任何时候翻转了小丑，游戏就会结束，你将一无所获。

你愿意玩这个游戏的意愿有多少？

I give you a shuffled deck of $14$ cards which contain exactly one card from each rank and a Joker. You win $100 if you flip over the King, Queen, & Jack but if at any point you have flipped over the Joker, the game ends and you win nothing.

How much would you be willing to play this game?

解析

唯一对游戏结果有影响的牌是国王、王后、杰克和小丑，因此只需要考虑这些 $4$ 卡。

如果小丑出现在这三张皇家卡中的任何一张之前，你就输了游戏，因此你获胜的概率与小丑在这些卡中最后被选中的概率相同。 $4$ 卡中的任何 $1$ 最后的概率是 $\frac14$ 。

你玩这款游戏的预期收入是 $\frac14\cdot100$ ，因此你应该愿意为这款游戏支付 $\boxed{25}$ 美元。

Original Explanation

The only cards that have an effect on the outcome of the game are the King, Queen, Jack, & Joker so only these $4$ cards need to be considered.

You lose the game if the Joker comes before any one of these three royal cards, so the probability of you winning is the same as the probability the Joker is selected last among these cards. The probability any $1$ of the $4$ cards is last is $\frac14$ .

Your expected earnings from playing this game is $\frac14\cdot100$ so you should be willing to pay up until $\boxed{25}$ dollars for this game.