扑克牌手牌概率

总手牌数：$\binom{52}{5}=2{,}598{,}960$。

1. 四条：选点数 $13$ 种，副牌从剩余 $48$ 张选 1：$13\cdot 48$。

$$P=\frac{13\cdot 48}{\binom{52}{5}}.$$

2. 葫芦：三条点数 $13$ 选 1 且花色 $\binom{4}{3}=4$；对子点数从剩余 12 选 1 且花色 $\binom{4}{2}=6$。

$$P=\frac{13\cdot 4\cdot 12\cdot 6}{\binom{52}{5}}.$$

3. 两对：选两种点数 $\binom{13}{2}$，每对选花色 $\binom{4}{2}^2$，再从剩余 44 张选 1 张单牌。

$$P=\frac{\binom{13}{2}\cdot \binom{4}{2}^2\cdot 44}{\binom{52}{5}}.$$

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英文解析

The total number of 5-card hands is

$$\binom{52}{5} \;=\; 2{,}598{,}960.$$

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.
     $$P = \frac{13 \times 48}{2{,}598{,}960} \;=\; \frac{624}{2{,}598{,}960}.$$

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

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

3. Two Pairs

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