【学习资料】袋中共36个球，红白黑格12个，问能一次抽到3个红4个白5个黑的概率是多少？

1、公式计算

1.1 题目1

袋中共$36$个球，$\boxed{红}$$\boxed{白}$$\boxed{黑}$各$12$个，问能一次抽到$\boxed{3个红}$$\boxed{4个白}$$\boxed{5个黑}$的概率是多少？

$$P_{m_r,m_w,m_b}=\frac{C_{n_r}^{m_r}\times C_{n_w}^{m_w}\times C_{n_b}^{m_b}}{C_n^m}$$
其中：
$P_{m_r,m_w,m_b}$ – 概率
$C_n^m$ – n个球中，抽m个球的组合数。
$C_{n_r}^{m_r}$ – $n_r$个红球中，抽$m_r$个红球的组合数。
$C_{n_w}^{m_w}$ – $n_w$个白球中，抽$m_r$个白球的组合数。
$C_{n_b}^{m_b}$ – $n_b$个黑球中，抽$m_b$个黑球的组合数。

根据已知条件代入公式：
$$已知条件\{\begin{array}{l}{n}_r=12\\ n_w=12\\ n_b=12\\ m_r=3\\ m_w=4\\ m_b=5\\ n=36\\ m=12\end{array}$$

$$\begin{array}{c}\begin{array}{rl}{P}_{3_r,4_w,5_b}& =\frac{C_{12}^3\times C_{12}^4\times C_{12}^5}{C_{36}^{12}}\\ & =\frac{\frac{12!}{3!\times (12-3)!}\times \frac{12!}{4!\times (12-4)!}\times \frac{12!}{5!\times (12-5)!}}{\frac{36!}{12!\times (36-12)!}}\\ & =\frac{\frac{12!}{3!\times 9!}\times \frac{12!}{4!\times 8!}\times \frac{12!}{5!\times 7!}}{\frac{36!}{12!\times 24!}}\\ & =\frac{\frac{12!}{3!\times (12-3)!}\times \frac{12!}{4!\times (12-4)!}\times \frac{12!}{5!\times (12-5)!}}{\frac{36!}{12!\times (36-12)!}}\\ & =\frac{\frac{479,001,600}{6\times 362,880}\times \frac{479,001,600}{24\times 40,320}\times \frac{479,001,600}{120\times 5,040}}{\frac{371,993,326,789,901,217,467,999,448,150,835,200,000,000}{479,001,600\times 620,448,401,733,239,439,360,000}}\\ & =\frac{220\times 495\times 792}{1,251,677,700}\\ & =\frac{86,248,800}{1,251,677,700}\\ & =\frac{95832}{1390753}\\ & =0.06890655637629399325401419231165\end{array}\end{array}$$

1.2 题目2

袋中共 $36$ 个球，$\boxed{红}$$\boxed{白}$$\boxed{黑}$各$12$个，问能一次抽到颜色组合是 $3$、$4$、$5$ 的概率是多少？

$$P_{m_1,m_2,m_3}=\frac{N!}{N_1!\times N_2!\times \dots \times N_k!}\times \frac{C_{n_1}^{m_1}\times C_{n_2}^{m_2}\times C_{n_3}^{m_3}}{C_n^m}$$
其中：
$P_{m_1,m_2,m_3}$ – 概率
$C_n^m$ – n个球中，抽m个球的组合数。
$C_{n_1}^{m_1}$ – $n_1$个颜色1球中，抽$m_1$个颜色1球的组合数。
$C_{n_2}^{m_2}$ – $n_2$个颜色2球中，抽$m_2$个颜色2球的组合数。
$C_{n_3}^{m_3}$ – $n_3$个颜色3球中，抽$m_3$个颜色3球的组合数。
$\frac{N!}{N_1!\times N_2!\times \dots \times N_k!}$ – $m_1、m_2、m_3$数的排列组合数。

$m_1、m_2、m_3$	$\frac{N!}{N_1!\times N_2!\times \dots \times N_k!}$	排列组合
3,4,5	第1个数3出现1次 第2个数4出现1次 第3个数5出现1次 $\frac{3!}{1!\times 1!\times 1!}$	3,4,5 3,5,4 4,3,5 4,5,3 5,4,3 5,3,4
4,4,4	第1个数4出现3次 $\frac{3!}{3!}$	4,4,4
3,3,6	第1个数3出现2次 第2个数6出现1次 $\frac{3!}{2!\times 1!}$	3,3,6 3,6,3 6,3,3
1,1,10	第1个数1出现2次 第2个数10出现1次 $\frac{3!}{2!\times 1!}$	1,1,10 1,10,1 10,1,1
$\dots$	$\dots$	$\dots$

根据已知条件代入公式：
$$已知条件\{\begin{array}{l}{n}_1=12\\ n_2=12\\ n_3=12\\ m_1=3\\ m_2=4\\ m_3=5\\ n=36\\ m=12\\ N=3\\ N_{1(3)}=1\\ N_{2(4)}=1\\ N_{3(5)}=1\end{array}$$

$$\begin{array}{c}\begin{array}{rl}{P}_{3,4,5}& =\frac{3!}{1!\times 1!\times 1!}\times \frac{C_{12}^3\times C_{12}^4\times C_{12}^5}{C_{36}^{12}}\\ & =\frac{6}{1}\times \frac{\frac{12!}{3!\times (12-3)!}\times \frac{12!}{4!\times (12-4)!}\times \frac{12!}{5!\times (12-5)!}}{\frac{36!}{12!\times (36-12)!}}\\ & =6\times \frac{\frac{12!}{3!\times 9!}\times \frac{12!}{4!\times 8!}\times \frac{12!}{5!\times 7!}}{\frac{36!}{12!\times 24!}}\\ & =6\times \frac{\frac{12!}{3!\times (12-3)!}\times \frac{12!}{4!\times (12-4)!}\times \frac{12!}{5!\times (12-5)!}}{\frac{36!}{12!\times (36-12)!}}\\ & =6\times \frac{\frac{479,001,600}{6\times 362,880}\times \frac{479,001,600}{24\times 40,320}\times \frac{479,001,600}{120\times 5,040}}{\frac{371,993,326,789,901,217,467,999,448,150,835,200,000,000}{479,001,600\times 620,448,401,733,239,439,360,000}}\\ & =6\times \frac{220\times 495\times 792}{1,251,677,700}\\ & =6\times \frac{86,248,800}{1,251,677,700}\\ & =6\times \frac{95832}{1390753}\\ & =\frac{574992}{1390753}\\ & =0.41343933825776395952408515386988\end{array}\end{array}$$

2、枚举法

使用程序模拟每一次抽取，计算每一步的概率，并进行累计汇总。

using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Threading.Tasks;
using static GL39_9.Program;namespace GL39_9
{internal class Program{/// <summary>/// 汇总类型/// </summary>public enum eSummaryType{/// <summary>/// 按数量汇总/// </summary>Number,/// <summary>/// 按名称和数量汇总/// P(36,12(3,4,5))=N! * C(12,3)*C(12,4)*C(12,5)/C(36,12)/// C(12,3) = 12!/(3!*(12-3)!) = 12!/(3!*9!)/// C(12,4) = 12!/(4!*(12-4)!) = 12!/(4!*8!)/// C(12,4) = 12!/(5!*(12-5)!) = 12!/(5!*7!)/// 3 4 5 (N=3)/// 4 4 4 (N=1)/// 3 3 6 (N=2)/// </summary>NameNumber}public static double Calc(int[] keys,int pre=12){int nType = keys.GroupBy(o=>o).Count();BigInteger FT = Factorial(nType);int t = pre * keys.Count();int c = keys.Sum(o=>o);BigInteger fz = 1;foreach (var item in keys){fz *= C(pre, item);}BigInteger fm = C(t,c);return ((double)(long)FT * ((double)(long)fz / (double)(long)fm));}public static BigInteger C(int n,int m){var fn = Factorial(n);var fm = Factorial(m);var fmn = Factorial(n-m);return fn /(fm * fmn);}public static BigInteger Factorial(int n){if (n < 0) throw new ArgumentException("n must be non-negative.");if (n == 0) return 1;return n * Factorial(n - 1);}/// <summary>/// 暗盒/袋中的物品信息/// </summary>public class CBlackBoxItem{/// <summary>/// 物品唯一编号/// </summary>public int Id { get; set; }/// <summary>/// 物品名称/// </summary>public string Name { get; set; }/// <summary>/// 物品数量/// </summary>public int Number { get; set; }/// <summary>/// /// </summary>/// <param name="id">物品唯一编号</param>/// <param name="name">物品名称</param>/// <param name="number">物品数量</param>public CBlackBoxItem(int id, string name, int number){Id = id;Name = name;Number = number;}/// <summary>/// 物品信息/// </summary>/// <returns></returns>public override string ToString(){return $"{Id}.{Name} {Number}个";}/// <summary>/// 复制物品信息及数量/// </summary>/// <returns></returns>public CBlackBoxItem Copy(){return new CBlackBoxItem(Id, Name, Number);}/// <summary>/// 复制物品并减去指定数量的物品/// </summary>/// <param name="val"></param>/// <returns></returns>public CBlackBoxItem CopySub(int val){return new CBlackBoxItem(Id, Name, Number - val);}/// <summary>/// 复制物品并设置数量/// </summary>/// <param name="val"></param>/// <returns></returns>public CBlackBoxItem Copy(int val){return new CBlackBoxItem(Id, Name, val);}}/// <summary>/// 轮次信息/// </summary>public class CRoundInformation{/// <summary>/// 前一轮的信息/// </summary>public CRoundInformation previousRound = null;/// <summary>/// 本轮暗箱信息/// </summary>public List<CBlackBoxItem> CurrentRoundBlackBox = new List<CBlackBoxItem> {new CBlackBoxItem(1,"红色球/棍子",9),new CBlackBoxItem(2,"绿色球/棍子",9),new CBlackBoxItem(3,"蓝色球/棍子",9),};/// <summary>/// 从暗盒中获取总数量/// </summary>/// <returns></returns>public int GetTotalNumberFromBlackBox(){return CurrentRoundBlackBox.Select(o => o.Number).Sum();}/// <summary>/// 当前从暗箱中抽中的物品/// </summary>public CBlackBoxItem CurrentDrawnBlackBoxItem = null;/// <summary>/// 当前轮次概率/// </summary>public double CurrentRoundProbability = 0;/// <summary>/// 累计所有轮的概率/// </summary>public double CumulativeProbability = 0;/// <summary>/// 累计所有抽到的物品/// </summary>public List<CBlackBoxItem> CumulativeDrawnBlackBoxItems = new List<CBlackBoxItem>();/// <summary>/// 所有抽到的物品归类信息/// </summary>public string SummaryInformation = "";/// <summary>/// 轮次/// </summary>public int RoundIndex = 0;/// <summary>/// 更新累计的概率/// </summary>public void UpdateCumulativeProbability(){double probability = 1.0;GetCumulativeProbability(this, ref probability);CumulativeProbability = probability;}/// <summary>/// 递归获取累计概率/// </summary>/// <param name="roundInformation"></param>/// <param name="probability"></param>private void GetCumulativeProbability(CRoundInformation roundInformation, ref double probability){if (roundInformation == null) return;if (roundInformation.RoundIndex <= 0) return;probability *= roundInformation.CurrentRoundProbability;GetCumulativeProbability(roundInformation.previousRound, ref probability);}/// <summary>/// 统计抽到的物品，获取汇总信息/// </summary>public void UpdateDrawnBlackBoxItems(eSummaryType summaryType = eSummaryType.Number){List<CBlackBoxItem> blackBoxItems = new List<CBlackBoxItem>();GetDrawnBlackBoxItems(this, ref blackBoxItems);CumulativeDrawnBlackBoxItems = blackBoxItems;List<CBlackBoxItem> temps = new List<CBlackBoxItem>();foreach (var item in CurrentRoundBlackBox){temps.Add(item.Copy(blackBoxItems.Where(o => o.Id == item.Id).Count()));}temps = temps.OrderByDescending(o => o.Number).ToList();if(summaryType == eSummaryType.Number){SummaryInformation = string.Join(":", temps.Select(o => o.Number));}else if (summaryType == eSummaryType.NameNumber){SummaryInformation = string.Join(":", temps.Select(o => o.Number)) +"|" + string.Join(":", temps);}}/// <summary>/// 递归获取每一轮的物品/// </summary>/// <param name="roundInformation"></param>/// <param name="blackBoxItems"></param>private void GetDrawnBlackBoxItems(CRoundInformation roundInformation, ref List<CBlackBoxItem> blackBoxItems){if (roundInformation == null) return;if (roundInformation.RoundIndex <= 0) return;blackBoxItems.Add(roundInformation.CurrentDrawnBlackBoxItem);GetDrawnBlackBoxItems(roundInformation.previousRound, ref blackBoxItems);}/// <summary>/// 从暗盒中抽一个物品/// </summary>/// <param name="roundIndex"></param>/// <returns></returns>public List<CRoundInformation> DrawnOneItemFromBlackBox(int roundIndex){List<CRoundInformation> roundInformations = new List<CRoundInformation>();var total = GetTotalNumberFromBlackBox();foreach (var item in CurrentRoundBlackBox){if (item.Number <= 0){continue;}CRoundInformation roundInformation = new CRoundInformation();roundInformation.RoundIndex = roundIndex;roundInformation.previousRound = this;roundInformation.CurrentDrawnBlackBoxItem = new CBlackBoxItem(item.Id, item.Name, 1);roundInformation.CurrentRoundProbability = item.Number / (double)total;roundInformation.CurrentRoundBlackBox.Clear();foreach (var itme in CurrentRoundBlackBox){if (item.Id == itme.Id){roundInformation.CurrentRoundBlackBox.Add(itme.CopySub(1));}else{roundInformation.CurrentRoundBlackBox.Add(itme.Copy());}}roundInformations.Add(roundInformation);}return roundInformations;}}/// <summary>/// 概率小游戏/// </summary>public class CProbabilityGame{public List<CRoundInformation> Source = new List<CRoundInformation>();public int DrawnTimes = 9;/// <summary>/// 每次抽取花费/// 100块抽5次/// </summary>public double SpendEveryTime = 20;public eSummaryType SummaryType =   eSummaryType.Number;/// <summary>/// 奖池/// </summary>public Dictionary<string, double> Jackpots = new Dictionary<string, double>{{"9:0:0",100},//:  0.000640092173272951‰{"8:1:0",30},//:  0.103694932070218‰{"7:2:0",10},//:  1.65911891312349‰{"7:1:1",20},//:  1.86650877726393‰{"6:3:0",10},//:  9.03298074922795‰{"6:2:1",10},//:  34.8414971755928‰{"5:4:0",2},//:  20.3242066857629‰{"5:3:1",2},//:  121.945240114572‰{"5:2:2",1},//:  104.524491526778‰{"4:4:1",1},//:  91.4589300859336‰{"4:3:2",-10},//:  487.780960458368‰{"3:3:3",1}//:  126.461730489187‰};public CProbabilityGame(){//Init312Drawn12();Init39Drawn9();}/// <summary>/// 3 种颜色的物品 每种 12个，抽12个/// </summary>public void Init312Drawn12(){DrawnTimes = 12;SummaryType = eSummaryType.Number;var item = new CRoundInformation();item.CurrentRoundBlackBox.Clear();item.CurrentRoundBlackBox = new List<CBlackBoxItem> {new CBlackBoxItem(1,"红色球/棍子",DrawnTimes),new CBlackBoxItem(2,"绿色球/棍子",DrawnTimes),new CBlackBoxItem(3,"蓝色球/棍子",DrawnTimes),};Source.Add(item);Jackpots.Clear();Jackpots = new Dictionary<string, double>{{"12:0:0",100},//2.39678313354948E-06‰{"11:1:0",30},//0.00069027354246225‰{"10:2:0",20},//0.020880774659483‰{"10:1:1",20},//0.0227790269012542‰{"9:3:0",10},//0.232008607327592‰{"9:2:1",10},//0.835230986379351‰{"8:4:0",5},//1.17454357459599‰{"8:3:1",5},//6.26423239784602‰{"8:2:2",5},//5.16799172822235‰{"7:5:0",5},//3.0068315509653‰{"7:4:1",2},//22.551236632241‰{"7:3:2",2},//55.1252451009996‰{"6:6:0",5},//2.04631591662927‰{"6:5:1",2},//42.0956417134937‰{"6:4:2",1},//144.70376839016‰{"6:3:3",1},//107.187976585368‰{"5:5:2",1},//99.2254411818329‰{"5:4:3",-10},//413.439338257532‰{"4:4:4",1},//96.8998449041164‰};}/// <summary>/// 3 种颜色的物品 每种 9个，抽9个/// </summary>public void Init39Drawn9(){DrawnTimes = 9;SpendEveryTime = 0.5;SummaryType = eSummaryType.Number;var item = new CRoundInformation();item.CurrentRoundBlackBox.Clear();item.CurrentRoundBlackBox = new List<CBlackBoxItem> {new CBlackBoxItem(1,"红色球/棍子",DrawnTimes),new CBlackBoxItem(2,"绿色球/棍子",DrawnTimes),new CBlackBoxItem(3,"蓝色球/棍子",DrawnTimes),};Source.Add(item);Jackpots.Clear();Jackpots = new Dictionary<string, double>{{"9:0:0",100},//:  0.000640092173272951‰{"8:1:0",30},//:  0.103694932070218‰{"7:2:0",10},//:  1.65911891312349‰{"7:1:1",20},//:  1.86650877726393‰{"6:3:0",10},//:  9.03298074922795‰{"6:2:1",10},//:  34.8414971755928‰{"5:4:0",2},//:  20.3242066857629‰{"5:3:1",2},//:  121.945240114572‰{"5:2:2",1},//:  104.524491526778‰{"4:4:1",1},//:  91.4589300859336‰{"4:3:2",-10},//:  487.780960458368‰{"3:3:3",1}//:  126.461730489187‰};}/// <summary>/// 计算/// </summary>/// <returns></returns>public double Calculate(){List<CRoundInformation> listTemp = new List<CRoundInformation>();Drawn(ref listTemp);Cumulative(ref listTemp);var pro = SummaryProbabilitys(listTemp);if(CheckProbabilitys(pro)){return CalculateProfitLoss(pro);}return double.MinValue;}/// <summary>/// 检查概率是否归一/// </summary>/// <param name="Probabilitys"></param>/// <returns></returns>public bool CheckProbabilitys(Dictionary<string, double> Probabilitys){var val = Probabilitys.Sum(o => o.Value);System.Diagnostics.Debug.WriteLine(val);if(val>1-0.0000001 && val<1+0.0000001){return true;}return false;}/// <summary>/// 累计/// </summary>/// <param name="listTemp"></param>private void Cumulative(ref List<CRoundInformation> listTemp){foreach (var item in listTemp){item.UpdateCumulativeProbability();item.UpdateDrawnBlackBoxItems(SummaryType);}}/// <summary>/// 抽取/// </summary>/// <param name="listTemp"></param>private void Drawn(ref List<CRoundInformation> listTemp){listTemp = Source;for (int i = 1; i <= DrawnTimes; i++){var temps = new List<CRoundInformation>();foreach (var item in listTemp){if (item.RoundIndex < i - 1){continue;}var list = item.DrawnOneItemFromBlackBox(i);temps.AddRange(list);}listTemp = temps;}}/// <summary>/// 汇总/// </summary>/// <param name="listTemp"></param>/// <returns></returns>private Dictionary<string, double> SummaryProbabilitys(List<CRoundInformation> listTemp){Dictionary<string, double> Probabilitys = new Dictionary<string, double>();System.Diagnostics.Debug.WriteLine($"|组合|概率");System.Diagnostics.Debug.WriteLine($"|--|--|");foreach (var item in listTemp.GroupBy(o => o.SummaryInformation).OrderByDescending(o => o.Key)){var glall = item.Sum(o => o.CumulativeProbability);Probabilitys.Add(item.Key, glall);System.Diagnostics.Debug.WriteLine($"|{item.Key}|{glall * 100}%");//System.Diagnostics.Debug.WriteLine($"{{\"{item.Key}\",{1.0/ glall}}},//{glall*1000}‰");}return Probabilitys;}/// <summary>/// 计算盈亏/// </summary>private double CalculateProfitLoss(Dictionary<string, double> Probabilitys){double revenue = -SpendEveryTime;foreach (var item in Jackpots){if (Probabilitys.ContainsKey(item.Key)){revenue += Probabilitys[item.Key] * item.Value;}}return revenue;}static void Main(string[] args){CProbabilityGame game = new CProbabilityGame();var item = Program.Calc(new int[3] {3,4,5 },12);System.Diagnostics.Debug.WriteLine(item);var val = 20 * game.Calculate();System.Diagnostics.Debug.WriteLine(val);}}}
}

3、拓展

街头玩法

每次费用 $10/20=0.5元$
根据概率，抽一次罚$3.7749845577769543元$
加上成本 $0.5$ ，每把亏$4.2749845577769543元$，
每次交费玩$20$次，一次缴费$4.2749845577769543\times 20=85.499691155539086元$

常见玩法概率表

3种颜色物品，每种9个，每次抽9个的概率

组合	概率
9:0:0	0.0000640092173272951%
8:1:0	0.0103694932070218%
7:2:0	0.165911891312349%
7:1:1	0.186650877726393%
6:3:0	0.903298074922795%
6:2:1	3.48414971755928%
5:4:0	2.03242066857629%
5:3:1	12.1945240114572%
5:2:2	10.4524491526778%
4:4:1	9.14589300859336%
4:3:2	48.7780960458368%
3:3:3	12.6461730489187%

3种颜色物品，每种12个，每次抽12个的概率

组合	概率
12:0:0	0.000000239678313354948%
11:1:0	0.000069027354246225%
10:2:0	0.0020880774659483%
10:1:1	0.00227790269012542%
9:3:0	0.0232008607327592%
9:2:1	0.0835230986379351%
8:4:0	0.117454357459599%
8:3:1	0.626423239784602%
8:2:2	0.516799172822235%
7:5:0	0.30068315509653%
7:4:1	2.2551236632241%
7:3:2	5.51252451009996%
6:6:0	0.204631591662927%
6:5:1	4.20956417134937%
6:4:2	14.470376839016%
6:3:3	10.7187976585368%
5:5:2	9.92254411818329%
5:4:3	41.3439338257532%
4:4:4	9.68998449041164%