主题：概率论基础R程序和勘误

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概率论基础勘误表

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桥牌竞技中的概率问题

桥牌是一种概率游戏，其叫牌体系和打牌路线的设计要以概率计算为基础，牌力与控制力的估算标准也要靠概率分析作科学论证，概率常识已经成了牌手们掌握胜

算的诀窍。

1.常见牌型概率：依据概率计算，常见牌型概率如下(R程序)：
自己手上的牌型分布计算：
思路：每个花色都有13张，依次抽取。
4-4-3-2 21.55%   4!*(13!*39!/52!)*(13!/4!/9!)*(13!/4!/9!)*(13!/3!/10!)*(13!/2!/11!)/2!
                 factorial(4)*choose(13,4)*choose(13,4)*choose(13,3)*choose(13,2)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
5-3-3-2 15.52%   factorial(4)*choose(13,5)*choose(13,3)*choose(13,3)*choose(13,2)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
5-4-3-1 12.93%   factorial(4)*choose(13,5)*choose(13,4)*choose(13,3)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)
5-4-2-2 10.58%   factorial(4)*choose(13,5)*choose(13,4)*choose(13,2)*choose(13,2)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
4-3-3-3 10.53%   factorial(4)*choose(13,4)*choose(13,3)*choose(13,3)*choose(13,3)*factorial(39)/factorial(52)*factorial(13)/factorial(3)
6-3-2-2  5.64%   factorial(4)*choose(13,6)*choose(13,3)*choose(13,2)*choose(13,2)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
6-4-2-1  4.70%   factorial(4)*choose(13,6)*choose(13,4)*choose(13,2)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)
6-3-3-1  3.45%   factorial(4)*choose(13,6)*choose(13,3)*choose(13,3)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
5-5-2-1  3.17%   factorial(4)*choose(13,5)*choose(13,5)*choose(13,2)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2) 
4-4-4-1  2.99%   factorial(4)*choose(13,4)*choose(13,4)*choose(13,4)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(3)
7-3-2-1  1.88%   factorial(4)*choose(13,7)*choose(13,3)*choose(13,2)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)
6-4-3-0  1.33%   factorial(4)*choose(13,6)*choose(13,4)*choose(13,3)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
5-4-4-0  1.24%   factorial(4)*choose(13,5)*choose(13,4)*choose(13,4)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
5-5-3-0  0.90%   factorial(4)*choose(13,5)*choose(13,5)*choose(13,3)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
6-5-1-1  0.71%   factorial(4)*choose(13,6)*choose(13,5)*choose(13,1)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
6-5-2-0  0.65%   factorial(4)*choose(13,6)*choose(13,5)*choose(13,2)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
7-2-2-2  0.51%   factorial(4)*choose(13,7)*choose(13,2)*choose(13,2)*choose(13,2)*factorial(39)/factorial(52)*factorial(13)/factorial(3)
7-4-1-1  0.39%   factorial(4)*choose(13,7)*choose(13,4)*choose(13,1)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
7-4-2-0  0.36%   factorial(4)*choose(13,7)*choose(13,4)*choose(13,2)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
7-3-3-0  0.27%   factorial(4)*choose(13,7)*choose(13,3)*choose(13,3)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
8-2-2-1  0.19%   factorial(4)*choose(13,8)*choose(13,2)*choose(13,2)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
8-3-1-1  0.12%   factorial(4)*choose(13,8)*choose(13,3)*choose(13,1)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
8-3-2-0  0.11%   factorial(4)*choose(13,8)*choose(13,3)*choose(13,2)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
7-5-1-0  0.11%   factorial(4)*choose(13,7)*choose(13,5)*choose(13,1)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
6-6-1-0  0.07%   factorial(4)*choose(13,6)*choose(13,6)*choose(13,1)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
8-4-1-0  0.05%   factorial(4)*choose(13,8)*choose(13,4)*choose(13,1)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
9-2-1-1  0.02%   factorial(4)*choose(13,9)*choose(13,2)*choose(13,1)*choose(13,1)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
7-6-0-0  0.01%   factorial(4)*choose(13,7)*choose(13,6)*choose(13,0)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)
9-2-2-0  0.01%   factorial(4)*choose(13,9)*choose(13,2)*choose(13,2)*choose(13,0)*factorial(39)/factorial(52)*factorial(13)/factorial(2)
其它     0.01%
某一个花色的分布计算：
4-4-3-2 21.55%  factorial(4)*choose(13,4)*choose(9,4)*choose(5,3)*factorial(39)/factorial(52)*(13*12*11*10)^2*(13*12)^2*11/factorial(2)
思路：四个方位全排列，由于4重复，需要除以2!，某花色13张牌，先抽4张放一方，再抽4张，抽3张，剩下2张。每一方最多13张牌，因此依次13*12*11...
殊途同归，结果是一样的。

2.防守方牌张分配概率：依据机会均等原则，防守方在某一门花色上的牌张分配似乎总是理当趋近于均匀，但桥牌分牌竟必不同于扔硬币，概率测算结果自然会有

些异常：牌张数=奇数(3,5,7)时，均匀分配(2-1,3-2,4-3)概率均大于50%，对庄家有利。牌张数=偶数(4,6,8)时，平均分配概率均小于50%，非均匀分配(3-1,4-

2,5-3)概率反而更大，对庄家不利。唯牌张数=2例外，1-1分配概率52%，2-0分配概率48%。依据概率公式计算的防守方牌张分配概率列表如下：
外面2张，1-1分布的概率52%  13/25(条件概率，第一张先选定，第二张在同一牌手手上只有12个位置供选择)
外面2张，2-0分布的概率48%  2*13/26*12/25
外面3张，2-1分布的概率是78% 2*choose(3,1)*13/26*12/25*13/24
外面3张，3-0分布的概率是22% 2*13/26*12/25*11/24
外面4张，2-2分布的概率只有40.7%，choose(4,2)*13/26*12/25*13/24*12/23将牌只有AK想要一墩不丢最好的路线是打3-1分布并考虑飞牌。
外面4张，3-1分布的概率是49.74% 2*choose(4,1)*13/26*12/25*11/24*13/23
外面4张，4-0分布的概率是9.57% 2*13/26*12/25*11/24*10/23
外面5张，3-2均匀分布的慨率有67.83% 2*choose(5,3)*13/26*12/25*11/24*13/23*12/22
外面6张，3-3分布的概率35.53% choose(6,3)*factorial(20)/factorial(26)*(13*12*11)^2
外面7张，4-3分布的概率62.18% 2*choose(7,3)*factorial(19)/factorial(26)*(13*12*11)^2*10
外面8张，4-4分布的概率32.72% choose(8,4)*factorial(18)/factorial(26)*(13*12*11*10)^2

 

兴趣就是动力，要开学了，本期上概率论，老是讲赌博起源乏味了，从桥牌入手吧，大量的牌型分布概率计算就很好，昨晚用R写了一个自动发牌程序，就遇见下面这副牌，如何叫牌都是个问题，南北有4?定约，东西有点力优势，如果东西有局，干扰的情况下如何进局?西逼叫情况下南北几乎肯定会叫上局去阻击，对东西牌手是巨大的考验。

 

此主题相关图片如下：桥牌案例1.png

R程序优化了一下，现在长这样，原来直接排序，10会排在最前面，A和Q顺序也不对，现在用了一个标签函数就解决了，下午玩了几付牌，这副牌南北的满贯很难叫，但叫上去必成。北家如果方块A变草花A，大满贯都必成，但更难叫了，那个黑桃10太关键了
此主题相关图片如下：@qo6dgy2jp}d9rj08oqmye.png

此主题相关图片如下：j`{w7wir@79wv$808uu9@a.png

#产生一付牌，不含大小王
pkp<-data.frame(id=c(1:52),hs=rep(c("S","D","H","C"),each=13),face=rep(c(2:10,"J","Q","K","A"),4))
#洗牌
random<-sample(c(1:52),size=52)
xpkp=pkp[random,]

#同时输出到文件

#发牌
N=xpkp[1:13,]
N1=N[order(N$id,decreasing=T),]
NC=N1[N1$hs %in% 'C',]$face
ND=N1[N1$hs %in% 'D',]$face
NH=N1[N1$hs %in% 'H',]$face
NS=N1[N1$hs %in% 'S',]$face
print(NS, quote = FALSE);print(NH, quote = FALSE);print(ND, quote = FALSE);print(NC, quote = FALSE)

S=xpkp[27:39,]
S1=S[order(S$id,decreasing=T),]
SC=S1[S1$hs %in% 'C',]$face
SD=S1[S1$hs %in% 'D',]$face
SH=S1[S1$hs %in% 'H',]$face
SS=S1[S1$hs %in% 'S',]$face
print(SS, quote = FALSE);print(SH, quote = FALSE);print(SD, quote = FALSE);print(SC, quote = FALSE)

 

#西方的牌
W=xpkp[40:52,]
W1=W[order(W$id,decreasing=T),]
WC=W1[W1$hs %in% 'C',]$face
WD=W1[W1$hs %in% 'D',]$face
WH=W1[W1$hs %in% 'H',]$face
WS=W1[W1$hs %in% 'S',]$face
print(WS, quote = FALSE);print(WH, quote = FALSE);print(WD, quote = FALSE);print(WC, quote = FALSE)
#东方的牌
E=xpkp[14:26,]
E1=E[order(E$id,decreasing=T),]
EC=E1[E1$hs %in% 'C',]$face
ED=E1[E1$hs %in% 'D',]$face
EH=E1[E1$hs %in% 'H',]$face
ES=E1[E1$hs %in% 'S',]$face
print(ES, quote = FALSE);print(EH, quote = FALSE);print(ED, quote = FALSE);print(EC, quote = FALSE)

#作图
library(plotrix)
plot(0:20,0:20,type="n",xlab="", ylab="",axes=F);
rect(8, 8, 12, 12.3)
text(10,11.5,expression(N),cex=1.5)

#东方
text(14,12,"?",cex=1);text(14.6:(13.6+length(ES)),12,ES,cex=1)
text(14,11,"?",cex=1,col="red");text(14.6:(13.6+length(EH)),11,EH,cex=1)
text(14,10,"?",cex=1,col="red");text(14.6:(13.6+length(ED)),10,ED,cex=1)
text(14,9,"?",cex=1);text(14.6:(13.6+length(EC)),9,EC,cex=1)

#北方
text(7.4,17,"?",cex=1);text(8:(7+length(NS)),17,NS,cex=1)
text(7.4,16,"?",cex=1,col="red");text(8:(7+length(NH)),16,NH,cex=1)
text(7.4,15,"?",cex=1,col="red");text(8:(7+length(ND)),15,ND,cex=1)
text(7.4,14,"?",cex=1);text(8:(7+length(NC)),14,NC,cex=1)
#南方
text(7.4,6.5,"?",cex=1);text(8:(7+length(SS)),6.5,SS,cex=1)
text(7.4,5.5,"?",cex=1,col="red");text(8:(7+length(SH)),5.5,SH,cex=1)
text(7.4,4.5,"?",cex=1,col="red");text(8:(7+length(SD)),4.5,SD,cex=1)
text(7.4,3.5,"?",cex=1);text(8:(7+length(SC)),3.5,SC,cex=1)

此主题相关图片如下：1.png