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赌场抽牌比大小

Casino game

专题
Probability / 概率
难度
L4

题目详情

赌场游戏：你抽 1 张牌，庄家也抽 1 张牌。若你的牌点数更大则你赢，否则庄家赢；若平局，也算庄家赢。问：你获胜的概率是多少？

Casino game: you draw 1 card, dealer draws 1, if your card is higher you win, else house wins. Tie => house wins. What is your probability of winning?

解析

共有 13 种点数。

平局概率为 113\frac{1}{13}。在非平局条件下（点数不同），你比庄家大与比庄家小对称，各占一半。

因此

P(赢)=12(1−113)=12⋅1213=613≈0.4615.P(\text{赢})=\frac{1}{2}\left(1-\frac{1}{13}\right)=\frac{1}{2}\cdot\frac{12}{13}=\frac{6}{13}\approx 0.4615.

Original Explanation

We have 13 ranks. Probability of tie is 113\tfrac{1}{13}. Then ignoring suits, you have a 50-50 chance to be higher or lower if no tie. So P(win)=12(1−113)=12×1213=613≈0.4615.P(\text{win}) = \tfrac12 \bigl(1 - \tfrac{1}{13}\bigr) = \tfrac12 \times \tfrac{12}{13} = \tfrac{6}{13} \approx 0.4615.