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【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？

article 2024/11/13 15:22:40

1、公式计算

1.1 题目1

袋中共 $36$ 个球， $\fcolorbox{red}{#FADADE}{\color{red}{红}}$ $\fcolorbox{white}{#808080}{\color{white}{白}}$ $\fcolorbox{#808080}{#0D0D0D}{\color{#808080}{黑}}$ 各 $12$ 个，问能一次抽到 $\fcolorbox{red}{#FADADE}{\color{red}{3个红}}$ $\fcolorbox{white}{#808080}{\color{white}{4个白}}$ $\fcolorbox{#808080}{#0D0D0D}{\color{#808080}{5个黑}}$ 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：
$\begin{cases} n_r=12 \\ n_w=12 \\ n_b=12 \\ m_r=3 \\ m_w=4 \\ m_b=5 \\ n=36 \\ m=12 \\ \end{cases}$

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

1.2 题目2

$P_{m_1,m_2,m_3} =\frac{N!}{N_1! \times N_2! \times \ldots \times N_k!} \times \frac{C_{n_1}^{m_1} \times C_{n_2}^{m_2} \times C_{n_3}^{m_3} }{C_{n}^{m}}$
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。
$\frac{N!}{N_1! \times N_2! \times \ldots \times N_k!}$ – $m_1、m_2、m_3$ 数的排列组合数。

$m_1、m_2、m_3$	$\frac{N!}{N_1! \times N_2! \times \ldots \times N_k!}$	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
$\ldots$	$\ldots$	$\ldots$

根据已知条件代入公式：
$\begin{cases} n_1=12 \\ n_2=12 \\ n_3=12 \\ m_1=3 \\ m_2=4 \\ m_3=5 \\ n=36 \\ m=12 \\ N=3\\ N_{1(3)}=1\\ N_{2(4)}=1\\ N_{3(5)}=1\\ \end{cases}$

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

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

在这里插入图片描述
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%