弃牌游戏
DisCard Game
题目详情
赌场用一副标准 52 张牌。每回合翻开两张牌（共 26 回合直到翻完）：
- 两张都为黑：放入庄家牌堆。
- 两张都为红：放入你的牌堆。
- 一红一黑：两张都丢弃。
若你的牌堆张数严格多于庄家，你赢 10；否则（包括平局）你得到 0。
问：该游戏的公平价值（期望收益）是多少？
英文原题
Your playing a card game in a casino using a normal deck of cards laid down. On each turn you turn over two cards each time, for each pair of cards, if they are both are black, they go to the dealer's pile; if both are red, they go to your pile; else, they are discarded. The process is repeated times until all cards are gone. If you have more cards in your pile, you win $10. Otherwise if you lose, including ties, you get nothing. What is the fair value of this game?
解析
设翻完后你得到的“红红对”数量为 ，庄家得到的“黑黑对”数量为 。你的牌数为 ，庄家牌数为 。
注意到一旦出现一对红黑被丢弃，就同时消耗 1 张红牌与 1 张黑牌；而剩余的红牌只能两两配对形成红红对，剩余黑牌也只能两两配对形成黑黑对。
因为整副牌红黑各 26 张，丢弃红黑对后剩余红牌数与剩余黑牌数仍然相等，因此必有 ，从而你的牌数永远不可能严格多于庄家。
所以你获胜概率为 0，期望收益为 0，公平价值为 0。
英文解析
Set the number of "red red pairs" to after the turnover, and the number of "black pairs" to the dealer is . You have cards and the dealer has cards.
Note that once a pair of red and black cards is discarded, one red card and one black card are consumed at the same time; the remaining red cards can only be paired to form a red-red pair, and the remaining black cards can only be paired to form a black-black pair.
Because the entire deck of cards is 26 red and black, the number of remaining red cards after discarding the red and black pair is still equal to the number of remaining black cards, so there must be , so your card number can never be strictly more than the dealer.
So your probability of winning is 0, the expected gain is 0, and the fair value is * * 0 * *.