游戏王概率查表

2023-06-18 补充概率表 1.5; 优化表格加载速度
2021-09-15 修复1.4公式错误(0A0B1+壶不成立)；补充概率表1.4
2021-09-12 表1.1嵌入1.3
2021-09-08 补充概率表 1.3
2021-08-22 补充概率表 1.1, 1.2; 修正1.2简化公式错误

最近有在吹逼游戏王相关，强互动的中速策略为主的环境比较能提起兴致，虽然新冠坐牢遥遥无期，但人还是要有梦想，没事做做环境分析。随风旅鸟这个策略从发售后就上位数寥寥，日本“权威”排表中稳定二线开外，必然是有其致命短板，第一直觉是系统安定性的问题（包含系统初动率和抗压两个维度），而初动率就达不到环境平均水平，抗压方面展开链条每个环节都吃无效系，没有单卡补点，只能依赖抹杀、月书这类系统外对抗牌，系统虽然不大（14张左右）但初动率补助依靠大量壶压缩卡组（系统能搭载所有种类壶是优点，但不得不投入就是缺点了，尤其对后手阻抗上手率毫无帮助反而挤占其卡位，已经不是Hoban的无脑3Upstart Goblin的时代了），系统外卡位相比环境其他中速策略并没有优势。就系统初动率的问题，台北提供了一个思路，压缩围绕场地的2卡combo相关组件，投入小世界现象补助知更单卡初动的思路，我觉得是有其先进性的。小世界能联结系统内和系统外形成一个新的卡片集群，作为新卡的可能性是尚待开发的。但小世界本质也是2卡combo，需要投入特定中介来扩展小世界的配对范围，有优化手牌结构的作用但cost1手的代价会影响其他战术目标的达成。光空口白话列举优劣是很难得出有效的结论的，还是需要将不同构筑投射到同一的测度空间进行直接比较，从点到面建构出一个更直观的认识。

1 概率表

先把一些常见的组合情况做个列举，后面案例分析时会引用。

1.1 起手单卡

D = deck size， $40 \leq D \leq 60$

H = hand size，先手5，后手6

前提

一个目标卡片门类投入n张

P[起手1+] =

$\frac{\sum_{i=1}^{\min(n,H)}\binom{n}{i} \cdot \binom{D -n}{H-i}}{\binom{D}{H}}$

= 1 - P[起手0] =

$1 - \frac{\binom{n}{0} \cdot \binom{D -n}{H}}{\binom{D}{H}}$

n	prob
1	12.50%
2	23.72%
3	33.76%
4	42.71%
5	50.66%
6	57.71%
7	63.93%
8	69.40%
9	74.18%
10	78.34%

1.2 起手2卡combo

前提

两个目标卡片门类（A和B），各投入n和m张

P[起手各1+] =

$\frac { {\color{blue}\sum}_{i=1}^{\min(n,H)} {\color{blue}(} \binom{n}{i} \cdot {\color{red}\sum}_{j=1}^{\min(m,H-i)} {\color{red}(} \binom{m}{j} \cdot \binom{D-n-m}{H-i-j} {\color{red})} {\color{blue})} } {\binom{D}{H}}$

= 1 - P[起手0A 1+B] - P[起手1+A 0B] - P[起手0A 0B]

暂不考虑 $(D-n-m) < H$ 的情况

n \ m	1	2	3	4	5	6	7	8	9	10
1	$\color{blue}{1.28\%}$	2.46%	3.55%	4.54%	5.45%	6.28%	7.04%	7.72%	8.34%	8.89%
2	$\color{blue}{2.46\%}$	$\color{blue}{4.73\%}$	6.81%	8.71%	10.45%	12.03%	13.47%	14.77%	15.94%	17.00%
3	3.55%	$\color{blue}{6.81\%}$	$\color{blue}{9.80\%}$	12.53%	15.02%	17.29%	19.34%	21.20%	22.87%	24.37%
4	4.54%	8.71%	$\color{blue}{12.53\%}$	$\color{blue}{16.02\%}$	19.19%	22.08%	24.69%	27.04%	29.15%	31.05%
5	5.45%	10.45%	15.02%	$\color{blue}{19.19\%}$	$\color{blue}{22.99\%}$	26.42%	29.53%	32.33%	34.84%	37.08%
6	6.28%	12.04%	17.29%	22.08%	$\color{blue}{26.43\%}$	$\color{blue}{30.36\%}$	33.91%	37.11%	39.97%	42.52%
7	7.04%	13.47%	19.35%	24.69%	29.53%	$\color{blue}{33.92\%}$	$\color{blue}{37.86\%}$	41.40%	44.57%	47.39%
8	7.73%	14.78%	21.21%	27.05%	32.34%	37.11%	$\color{blue}{41.41\%}$	$\color{blue}{45.26\%}$	48.70%	51.75%
9	8.35%	15.96%	22.89%	29.17%	34.86%	39.98%	44.59%	$\color{blue}{48.71\%}$	$\color{blue}{52.38\%}$	55.63%
10	8.93%	17.04%	24.41%	31.09%	37.12%	42.55%	47.43%	51.78%	$\color{blue}{55.65\%}$	$\color{blue}{59.08\%}$

在同样卡位(n + m)的情况下，尽可能选择对角线上的组合( $| n - m | \leq 1$ )来最大化概率

1.3 壶补助单卡起手

前提

目标卡片门类投入n张
壶投入u张
- 效果翻/抽V张

P[前H张中0目标卡1+壶，后V张中1+目标卡] =

$\frac { \binom{n}{0} \cdot {\color{blue}\sum}_{i=1}^{\min(u,H)} {\color{blue}(} \binom{u}{i} \cdot \binom{D -n-u}{H-i} \cdot {\color{red}\sum}_{j=1}^{\min(n,V)} {\color{red}(} \binom{n}{j} \cdot \binom{D-H-n}{V-j} {\color{red})} {\color{blue})} } {\binom{D}{H+V} \cdot \binom{H+V}{H} = \binom{D}{H} \cdot \binom{D-H}{V}}$

底数两种算法是一样的，所以用等号标注
- 取(H+V)张，选其中H张为前H张，剩下自然为后V张
- 先取H张，然后从剩余的(D-H)张中取V张
暂时想不出简化公式（懒

1.3.1 强金/强贪 (v=2)

n \ u	0	1	2	3
1	12.50%	+0.64%	+1.21%	+1.73%
2	23.72%	+1.13%	+2.14%	+3.03%
3	33.76%	+1.49%	+2.81%	+3.99%
4	42.71%	+1.74%	+3.28%	+4.64%
5	50.66%	+1.90%	+3.57%	+5.04%
6	57.71%	+1.98%	+3.71%	+5.23%
7	63.93%	+1.99%	+3.74%	+5.26%
8	69.40%	+1.96%	+3.67%	+5.15%
9	74.18%	+1.89%	+3.53%	+4.94%
10	78.34%	+1.79%	+3.33%	+4.65%

1.3.2 强谦/金谦 (v=3)

n \ u	0	1	2	3
1	12.50%	+0.96%	+1.82%	+2.59%
2	23.72%	+1.67%	+3.16%	+4.48%
3	33.76%	+2.17%	+4.09%	+5.80%
4	42.71%	+2.49%	+4.70%	+6.65%
5	50.66%	+2.68%	+5.04%	+7.11%
6	57.71%	+2.75%	+5.16%	+7.27%
7	63.93%	+2.73%	+5.12%	+7.20%
8	69.40%	+2.64%	+4.95%	+6.94%
9	74.18%	+2.51%	+4.69%	+6.56%
10	78.34%	+2.34%	+4.36%	+6.09%

1.3.3 金谦 (v=6)

n \ u	0	1	2	3
1	12.50%	+1.92%	+3.64%	+5.18%
2	23.72%	+3.19%	+6.03%	+8.56%
3	33.76%	+3.95%	+7.47%	+10.58%
4	42.71%	+4.35%	+8.20%	+11.60%
5	50.66%	+4.47%	+8.41%	+11.88%
6	57.71%	+4.40%	+8.26%	+11.65%
7	63.93%	+4.20%	+7.87%	+11.07%
8	69.40%	+3.91%	+7.32%	+10.27%
9	74.18%	+3.57%	+6.67%	+9.34%
10	78.34%	+3.22%	+5.99%	+8.36%

真滴强，3-3就能达到10%的补助，注意这不光是对单一战术目标的提升，而是全局buff
壶的共同弱点，后攻无法补助手坑上手率，但可以补助解场牌
目标卡投入量超过5时，补助效率开始衰减，要结合表1.1来看

1.4 壶补助2卡combo起手

前提

两个目标卡片门类（A和B），各投入n和m张
壶投入u张
- 效果翻/抽V张

P[前H张中0A 1+B 1+壶，后V张中1+A] + P[前H张中1+A 0B 1+壶，后V张中1+B] =

$\frac { \binom{n}{0} \cdot {\color{blue}\sum}_{i=1}^{\min(u,H)} {\color{blue}(} \binom{u}{i} \cdot {\color{red}\sum}_{j=1}^{\min(m,H-i)} {\color{red}(} \binom{m}{j} \cdot \binom{D-n-m-u}{H-i-j} \cdot {\color{green}\sum}_{k=1}^{\min(n,V)} {\color{green}(} \binom{n}{k} \cdot \binom{D-H-n}{V-k} {\color{green})} {\color{red})} {\color{blue})} } {\binom{D}{H+V} \cdot \binom{H+V}{H} = \binom{D}{H} \cdot \binom{D-H}{V}} \\ + \frac { \binom{m}{0} \cdot {\color{blue}\sum}_{i=1}^{\min(u,H)} {\color{blue}(} \binom{u}{i} \cdot {\color{red}\sum}_{j=1}^{\min(n,H-i)} {\color{red}(} \binom{n}{j} \cdot \binom{D-n-m-u}{H-i-j} \cdot {\color{green}\sum}_{k=1}^{\min(m,V)} {\color{green}(} \binom{m}{k} \cdot \binom{D-H-m}{V-k} {\color{green})} {\color{red})} {\color{blue})} } {\binom{D}{H+V} \cdot \binom{H+V}{H} = \binom{D}{H} \cdot \binom{D-H}{V}}$

两项仅仅交换了n和m的位置
暂时想不出简化公式（懒

1.4.1 强金/强贪 (v=2)

1.4.1.1 投入1枚 (u=1)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.13%	+0.25%	+0.35%	+0.44%	+0.51%	+0.57%	+0.62%	+0.66%	+0.69%	+0.71%
2	+0.25%	+0.47%	+0.65%	+0.81%	+0.94%	+1.05%	+1.14%	+1.20%	+1.26%	+1.30%
3	+0.35%	+0.65%	+0.91%	+1.12%	+1.30%	+1.45%	+1.57%	+1.66%	+1.73%	+1.77%
4	+0.44%	+0.81%	+1.12%	+1.39%	+1.60%	+1.78%	+1.92%	+2.02%	+2.10%	+2.15%
5	+0.51%	+0.94%	+1.30%	+1.60%	+1.85%	+2.05%	+2.20%	+2.31%	+2.40%	+2.45%
6	+0.57%	+1.05%	+1.45%	+1.78%	+2.05%	+2.26%	+2.42%	+2.54%	+2.62%	+2.67%
7	+0.62%	+1.14%	+1.57%	+1.92%	+2.20%	+2.42%	+2.58%	+2.70%	+2.78%	+2.83%
8	+0.66%	+1.20%	+1.66%	+2.02%	+2.31%	+2.54%	+2.70%	+2.82%	+2.89%	+2.93%
9	+0.69%	+1.26%	+1.73%	+2.10%	+2.40%	+2.62%	+2.78%	+2.89%	+2.95%	+2.98%
10	+0.71%	+1.30%	+1.77%	+2.15%	+2.45%	+2.67%	+2.83%	+2.93%	+2.98%	+3.00%

1.4.1.2 投入2枚 (u=2)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.26%	+0.48%	+0.67%	+0.84%	+0.97%	+1.09%	+1.18%	+1.25%	+1.31%	+1.35%
2	+0.48%	+0.90%	+1.25%	+1.55%	+1.80%	+2.00%	+2.16%	+2.29%	+2.39%	+2.47%
3	+0.67%	+1.25%	+1.74%	+2.15%	+2.49%	+2.76%	+2.98%	+3.15%	+3.28%	+3.37%
4	+0.84%	+1.55%	+2.15%	+2.65%	+3.06%	+3.39%	+3.65%	+3.84%	+3.99%	+4.08%
5	+0.97%	+1.80%	+2.49%	+3.06%	+3.52%	+3.89%	+4.18%	+4.39%	+4.54%	+4.64%
6	+1.09%	+2.00%	+2.76%	+3.39%	+3.89%	+4.28%	+4.59%	+4.81%	+4.96%	+5.04%
7	+1.18%	+2.16%	+2.98%	+3.65%	+4.18%	+4.59%	+4.89%	+5.11%	+5.25%	+5.33%
8	+1.25%	+2.29%	+3.15%	+3.84%	+4.39%	+4.81%	+5.11%	+5.32%	+5.45%	+5.51%
9	+1.31%	+2.39%	+3.28%	+3.99%	+4.54%	+4.96%	+5.25%	+5.45%	+5.56%	+5.60%
10	+1.35%	+2.47%	+3.37%	+4.08%	+4.64%	+5.04%	+5.33%	+5.51%	+5.60%	+5.62%

1.4.1.3 投入3枚 (u=3)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.37%	+0.69%	+0.97%	+1.20%	+1.39%	+1.56%	+1.69%	+1.79%	+1.87%	+1.93%
2	+0.69%	+1.29%	+1.79%	+2.22%	+2.57%	+2.86%	+3.09%	+3.28%	+3.41%	+3.51%
3	+0.97%	+1.79%	+2.49%	+3.08%	+3.56%	+3.95%	+4.26%	+4.49%	+4.67%	+4.79%
4	+1.20%	+2.22%	+3.08%	+3.79%	+4.37%	+4.83%	+5.20%	+5.47%	+5.67%	+5.80%
5	+1.39%	+2.57%	+3.56%	+4.37%	+5.02%	+5.55%	+5.95%	+6.24%	+6.45%	+6.58%
6	+1.56%	+2.86%	+3.95%	+4.83%	+5.55%	+6.10%	+6.52%	+6.83%	+7.03%	+7.15%
7	+1.69%	+3.09%	+4.26%	+5.20%	+5.95%	+6.52%	+6.95%	+7.25%	+7.44%	+7.54%
8	+1.79%	+3.28%	+4.49%	+5.47%	+6.24%	+6.83%	+7.25%	+7.54%	+7.71%	+7.78%
9	+1.87%	+3.41%	+4.67%	+5.67%	+6.45%	+7.03%	+7.44%	+7.71%	+7.85%	+7.89%
10	+1.93%	+3.51%	+4.79%	+5.80%	+6.58%	+7.15%	+7.54%	+7.78%	+7.89%	+7.91%

1.4.2 强谦/金谦 (v=3)

1.4.2.1 投入1枚 (u=1)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.20%	+0.37%	+0.52%	+0.64%	+0.74%	+0.83%	+0.89%	+0.95%	+0.99%	+1.02%
2	+0.37%	+0.69%	+0.96%	+1.18%	+1.36%	+1.51%	+1.63%	+1.72%	+1.79%	+1.84%
3	+0.52%	+0.96%	+1.32%	+1.63%	+1.87%	+2.07%	+2.22%	+2.34%	+2.42%	+2.49%
4	+0.64%	+1.18%	+1.63%	+1.99%	+2.28%	+2.51%	+2.69%	+2.83%	+2.92%	+2.98%
5	+0.74%	+1.36%	+1.87%	+2.28%	+2.61%	+2.87%	+3.06%	+3.20%	+3.30%	+3.36%
6	+0.83%	+1.51%	+2.07%	+2.51%	+2.87%	+3.14%	+3.34%	+3.48%	+3.57%	+3.62%
7	+0.89%	+1.63%	+2.22%	+2.69%	+3.06%	+3.34%	+3.54%	+3.67%	+3.76%	+3.79%
8	+0.95%	+1.72%	+2.34%	+2.83%	+3.20%	+3.48%	+3.67%	+3.80%	+3.87%	+3.89%
9	+0.99%	+1.79%	+2.42%	+2.92%	+3.30%	+3.57%	+3.76%	+3.87%	+3.92%	+3.93%
10	+1.02%	+1.84%	+2.49%	+2.98%	+3.36%	+3.62%	+3.79%	+3.89%	+3.93%	+3.92%

1.4.2.2 投入2枚 (u=2)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.39%	+0.72%	+1.00%	+1.23%	+1.42%	+1.58%	+1.70%	+1.80%	+1.88%	+1.94%
2	+0.72%	+1.33%	+1.83%	+2.26%	+2.60%	+2.88%	+3.10%	+3.27%	+3.40%	+3.49%
3	+1.00%	+1.83%	+2.53%	+3.10%	+3.57%	+3.94%	+4.23%	+4.45%	+4.61%	+4.72%
4	+1.23%	+2.26%	+3.10%	+3.80%	+4.35%	+4.79%	+5.12%	+5.37%	+5.54%	+5.66%
5	+1.42%	+2.60%	+3.57%	+4.35%	+4.97%	+5.45%	+5.81%	+6.07%	+6.25%	+6.35%
6	+1.58%	+2.88%	+3.94%	+4.79%	+5.45%	+5.96%	+6.33%	+6.59%	+6.75%	+6.84%
7	+1.70%	+3.10%	+4.23%	+5.12%	+5.81%	+6.33%	+6.70%	+6.95%	+7.10%	+7.16%
8	+1.80%	+3.27%	+4.45%	+5.37%	+6.07%	+6.59%	+6.95%	+7.18%	+7.30%	+7.33%
9	+1.88%	+3.40%	+4.61%	+5.54%	+6.25%	+6.75%	+7.10%	+7.30%	+7.39%	+7.39%
10	+1.94%	+3.49%	+4.72%	+5.66%	+6.35%	+6.84%	+7.16%	+7.33%	+7.39%	+7.35%

1.4.2.3 投入3枚 (u=3)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.56%	+1.03%	+1.43%	+1.76%	+2.04%	+2.26%	+2.44%	+2.58%	+2.69%	+2.77%
2	+1.03%	+1.90%	+2.63%	+3.23%	+3.73%	+4.12%	+4.43%	+4.67%	+4.85%	+4.98%
3	+1.43%	+2.63%	+3.63%	+4.45%	+5.11%	+5.63%	+6.04%	+6.34%	+6.57%	+6.72%
4	+1.76%	+3.23%	+4.45%	+5.43%	+6.22%	+6.83%	+7.30%	+7.65%	+7.89%	+8.04%
5	+2.04%	+3.73%	+5.11%	+6.22%	+7.09%	+7.77%	+8.28%	+8.64%	+8.88%	+9.01%
6	+2.26%	+4.12%	+5.63%	+6.83%	+7.77%	+8.48%	+9.00%	+9.36%	+9.58%	+9.69%
7	+2.44%	+4.43%	+6.04%	+7.30%	+8.28%	+9.00%	+9.52%	+9.86%	+10.05%	+10.12%
8	+2.58%	+4.67%	+6.34%	+7.65%	+8.64%	+9.36%	+9.86%	+10.17%	+10.32%	+10.35%
9	+2.69%	+4.85%	+6.57%	+7.89%	+8.88%	+9.58%	+10.05%	+10.32%	+10.43%	+10.41%
10	+2.77%	+4.98%	+6.72%	+8.04%	+9.01%	+9.69%	+10.12%	+10.35%	+10.41%	+10.34%

1.4.3 金谦 (v=6)

1.4.3.1 投入1枚 (u=1)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.40%	+0.73%	+1.00%	+1.21%	+1.38%	+1.52%	+1.63%	+1.71%	+1.78%	+1.83%
2	+0.73%	+1.32%	+1.79%	+2.16%	+2.46%	+2.69%	+2.86%	+3.00%	+3.10%	+3.18%
3	+1.00%	+1.79%	+2.41%	+2.90%	+3.28%	+3.57%	+3.78%	+3.95%	+4.06%	+4.14%
4	+1.21%	+2.16%	+2.90%	+3.47%	+3.90%	+4.22%	+4.45%	+4.62%	+4.73%	+4.80%
5	+1.38%	+2.46%	+3.28%	+3.90%	+4.36%	+4.69%	+4.93%	+5.08%	+5.17%	+5.22%
6	+1.52%	+2.69%	+3.57%	+4.22%	+4.69%	+5.02%	+5.24%	+5.38%	+5.44%	+5.46%
7	+1.63%	+2.86%	+3.78%	+4.45%	+4.93%	+5.24%	+5.44%	+5.55%	+5.58%	+5.57%
8	+1.71%	+3.00%	+3.95%	+4.62%	+5.08%	+5.38%	+5.55%	+5.62%	+5.62%	+5.57%
9	+1.78%	+3.10%	+4.06%	+4.73%	+5.17%	+5.44%	+5.58%	+5.62%	+5.59%	+5.50%
10	+1.83%	+3.18%	+4.14%	+4.80%	+5.22%	+5.46%	+5.57%	+5.57%	+5.50%	+5.38%

1.4.3.2 投入2枚 (u=2)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+0.78%	+1.41%	+1.91%	+2.32%	+2.64%	+2.90%	+3.11%	+3.27%	+3.40%	+3.49%
2	+1.41%	+2.53%	+3.43%	+4.14%	+4.69%	+5.13%	+5.46%	+5.72%	+5.91%	+6.05%
3	+1.91%	+3.43%	+4.62%	+5.54%	+6.26%	+6.80%	+7.21%	+7.51%	+7.72%	+7.87%
4	+2.32%	+4.14%	+5.54%	+6.62%	+7.43%	+8.04%	+8.47%	+8.78%	+8.98%	+9.10%
5	+2.64%	+4.69%	+6.26%	+7.43%	+8.30%	+8.93%	+9.36%	+9.64%	+9.81%	+9.89%
6	+2.90%	+5.13%	+6.80%	+8.04%	+8.93%	+9.54%	+9.95%	+10.19%	+10.30%	+10.32%
7	+3.11%	+5.46%	+7.21%	+8.47%	+9.36%	+9.95%	+10.30%	+10.49%	+10.54%	+10.50%
8	+3.27%	+5.72%	+7.51%	+8.78%	+9.64%	+10.19%	+10.49%	+10.61%	+10.60%	+10.49%
9	+3.40%	+5.91%	+7.72%	+8.98%	+9.81%	+10.30%	+10.54%	+10.60%	+10.52%	+10.34%
10	+3.49%	+6.05%	+7.87%	+9.10%	+9.89%	+10.32%	+10.50%	+10.49%	+10.34%	+10.10%

1.4.3.3 投入3枚 (u=3)

n \ m	1	2	3	4	5	6	7	8	9	10
1	+1.12%	+2.02%	+2.75%	+3.33%	+3.79%	+4.16%	+4.45%	+4.68%	+4.85%	+4.99%
2	+2.02%	+3.64%	+4.92%	+5.93%	+6.72%	+7.34%	+7.81%	+8.17%	+8.44%	+8.63%
3	+2.75%	+4.92%	+6.62%	+7.94%	+8.95%	+9.72%	+10.29%	+10.71%	+11.00%	+11.20%
4	+3.33%	+5.93%	+7.94%	+9.47%	+10.62%	+11.47%	+12.08%	+12.50%	+12.78%	+12.94%
5	+3.79%	+6.72%	+8.95%	+10.62%	+11.85%	+12.73%	+13.33%	+13.71%	+13.93%	+14.03%
6	+4.16%	+7.34%	+9.72%	+11.47%	+12.73%	+13.59%	+14.14%	+14.47%	+14.61%	+14.63%
7	+4.45%	+7.81%	+10.29%	+12.08%	+13.33%	+14.14%	+14.63%	+14.88%	+14.93%	+14.86%
8	+4.68%	+8.17%	+10.71%	+12.50%	+13.71%	+14.47%	+14.88%	+15.03%	+14.99%	+14.81%
9	+4.85%	+8.44%	+11.00%	+12.78%	+13.93%	+14.61%	+14.93%	+14.99%	+14.85%	+14.57%
10	+4.99%	+8.63%	+11.20%	+12.94%	+14.03%	+14.63%	+14.86%	+14.81%	+14.57%	+14.21%

1.5 满编卡上手复数

前提

有n种类卡片满编投入（3张）

P[前H张中任意种类上手2或3的概率] =

$\frac{\sum_{i=2}^3{\binom{n}{1} \cdot \binom{3}{i} \cdot \binom{D - 3}{H - i} }}{\binom{D}{H}}$

n \ H	5 (先手)	6 (后手)
1	3.64%	5.36%
2	7.29%	10.73%
3	10.93%	16.09%
4	14.57%	21.46%
5	18.22%	26.82%
6	21.86%	32.19%
7	25.51%	37.55%
8	29.15%	42.91%
9	32.79%	48.28%
10	36.44%	53.64%

最后编辑于：2023.06.19 11:54:12

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