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扑克牌手牌概率：四条/葫芦/两对

Poker Hands

专题
Probability / 概率
难度
L4

题目详情

一副标准 52 张牌中随机抽取 5 张作为手牌。

求以下牌型的概率：

  1. 四条；
  2. 葫芦（3+2）；
  3. 两对。

In a standard 52-card deck (13 values ×\times 4 suits), a 5-card poker hand may have special configurations. What are the probabilities of getting:

  1. Four of a kind,
  2. A full house (three of a kind + a pair),
  3. Two pairs?
解析

总手牌数：(525)=2,598,960\binom{52}{5}=2{,}598{,}960。

  1. 四条：选点数 1313 种，副牌从剩余 4848 张选 1：13⋅4813\cdot 48。
P=13⋅48(525).P=\frac{13\cdot 48}{\binom{52}{5}}.
  1. 葫芦：三条点数 1313 选 1 且花色 (43)=4\binom{4}{3}=4；对子点数从剩余 12 选 1 且花色 (42)=6\binom{4}{2}=6。
P=13⋅4⋅12⋅6(525).P=\frac{13\cdot 4\cdot 12\cdot 6}{\binom{52}{5}}.
  1. 两对：选两种点数 (132)\binom{13}{2}，每对选花色 (42)2\binom{4}{2}^2，再从剩余 44 张选 1 张单牌。
P=(132)⋅(42)2⋅44(525).P=\frac{\binom{13}{2}\cdot \binom{4}{2}^2\cdot 44}{\binom{52}{5}}.

Original Explanation

The total number of 5-card hands is

(525)  =  2,598,960.\binom{52}{5} \;=\; 2{,}598{,}960.

  1. Four of a Kind

    • Choose 1 value out of 13 for the 4 cards,
    • Choose 1 card from the remaining 48 for the 5th card. P=13×482,598,960  =  6242,598,960.P = \frac{13 \times 48}{2{,}598{,}960} \;=\; \frac{624}{2{,}598{,}960}.

  2. Full House (3 cards of one value + 2 cards of another value)

    • Choose 1 value out of 13 for the triple ((43)=4\binom{4}{3}=4 suits),
    • Choose 1 value out of the remaining 12 for the pair ((42)=6\binom{4}{2}=6 suits). P=13×4×12×62,598,960  =  3,7442,598,960.P = \frac{13 \times 4 \times 12 \times 6}{2{,}598{,}960} \;=\; \frac{3{,}744}{2{,}598{,}960}.

  3. Two Pairs

    • Choose 2 values out of 13 for the pairs ((132)=78\binom{13}{2}=78),
    • For each pair, choose suits in (42)=6\binom{4}{2}=6 ways, so total 6×66 \times 6 for two pairs,
    • Choose 1 card from the remaining 44 cards. P=78×6×6×442,598,960  =  123,5522,598,960.P = \frac{78 \times 6 \times 6 \times 44}{2{,}598{,}960} \;=\; \frac{123{,}552}{2{,}598{,}960}.