投掷 N 个骰子获得所有可能值的概率

给定一个表示骰子数量的整数N ，任务是找到通过将N 个骰子放在一起可以获得的每个可能值的概率。

例子：

Input: N = 1 
Output: 
1: 0.17 
2: 0.17 
3: 0.17 
4: 0.17 
5: 0.17 
6: 0.17 
Explanation: On throwing a dice, the probability of all values from [1, 6] to appear at the top is 1/6 = 0.17

Input: N = 2 
Output: 
2: 0.028 
3: 0.056 
4: 0.083 
5: 0.11 
6: 0.14 
7: 0.17 
8: 0.14 
9: 0.11 
10: 0.083 
11: 0.056 
12: 0.028 
Explanation: The possible values of the sum of the two numbers that appear at the top on throwing two dices together ranges between [2, 12]. 

方法：思路是使用动态规划和DP表来存储每个可能值的概率。

• 存储在投掷 1 个骰子时可能出现的所有 6 个数字的概率。
• 现在，对于 N=2，[2, 12] 之间所有可能总和的概率等于两个数字相加后的相应概率的乘积之和。例如，

Probability of 4 on throwing 2 dices = (Probability of 1 ) * ( Probability of 3) + (Probability of 2) * ( Probability of 2) + (Probability of 3 ) * ( Probability of 1) 
 

• 因此对于 N 个骰子，

Probability of Sum S = (Probability of 1) * (Probability of S – 1 using N -1 dices) + (Probability of 2) * (Probability of S – 2 using N-1 dices) + ….. + (Probability of 6) * (Probability of S – 6 using N -1 dices) 
 

• 因此，为了解决这个问题，我们需要使用自顶向下的方法将 dp[][] 表从 2 填充到 N：

dp[i][x] = dp[1][y] + dp[i-1][z] where x = y + z and i denotes the number of dices 
 

• 显示为 N 存储的所有概率作为答案。

下面是上述方法的实现：

C++

// C++ Program to calculate
// the probability of
// all the possible values
// that can be obtained
// throwing N dices
 
#include 
using namespace std;
 
void dicesSum(int n)
{
    // Store the probabilities
    vector > dp(n + 1);
    // Precompute the probabilities
    // for values possible using 1 dice
    dp[1] = { { 1, 1 / 6.0 },
              { 2, 1 / 6.0 },
              { 3, 1 / 6.0 },
              { 4, 1 / 6.0 },
              { 5, 1 / 6.0 },
              { 6, 1 / 6.0 } };
 
    // Compute the probabilies
    // for all values from 2 to N
    for (int i = 2; i <= n; i++) {
        for (auto a1 : dp[i - 1]) {
            for (auto a2 : dp[1]) {
                dp[i][a1.first + a2.first]
                    += a1.second * a2.second;
            }
        }
    }
    // Print the result
    for (auto a : dp[n]) {
        cout << a.first << " "
             << setprecision(2)
             << a.second
             << endl;
    }
}
 
// Driver code
int main()
{
    int n = 2;
    dicesSum(n);
 
    return 0;
}

Java

// Java program to calculate
// the probability of all the
// possible values that can
// be obtained throwing N dices
import java.io.*;
import java.util.*;
 
class GFG{
 
static void dicesSum(int n)
{
     
    // Store the probabilities
    double[][] dp = new double[n + 1][6 * n + 1];
 
    // Precompute the probabilities
    // for values possible using 1 dice
    for(int i = 1; i <= 6; i++)
        dp[1][i] = 1 / 6.0;
 
    // Compute the probabilies
    // for all values from 2 to N
    for(int i = 2; i <= n; i++)
        for(int j = i - 1; j <= 6 * (i - 1); j++)
            for(int k = 1; k <= 6; k++)
            {
                dp[i][j + k] += (dp[i - 1][j] *
                                 dp[1][k]);
            }
 
    // Print the result
    for(int i = n; i <= 6 * n; i++)
    {
        System.out.println(i + " " +
                           Math.round(dp[n][i] * 1000.0) /
                                                 1000.0);
    }
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 2;
     
    dicesSum(n);
}
}
 
// This code is contributed by jithin

Python3

# Python3 program to calculate
# the probability of all the
# possible values that can
# be obtained throwing N dices
def diceSum(n):
     
    # Inititalize a 2d array upto
    # (n*total sum possible) sum
    # with value 0
    dp = [[ 0 for j in range(n * 6)]
              for i in range(n + 1)]
                   
    # Store the probability in a
    # single throw for 1,2,3,4,5,6
    for i in range(6):
        dp[1][i] = 1 / 6
         
    # Compute the probabilies
    # for all values from 2 to N
    for i in range(2, n + 1):
        for j in range(len(dp[i - 1])):
            for k in range(6):
                     
                if (dp[i - 1][j] != 0 and
                    dp[i - 1][k] != 0):
                    dp[i][j + k] += (dp[i - 1][j] *
                                     dp[1][k])
     
    # Print the result
    for i in range(len(dp[n]) - n + 1):
        print("%d %0.3f" % (i + n, dp[n][i]))
 
# Driver code
n = 2
 
# Call the function
diceSum(n)
 
# This code is contributed by dipesh99kumar

C#

// C# program to calculate
// the probability of all the
// possible values that can
// be obtained throwing N dices
using System;
class GFG {
     
    static void dicesSum(int n)
    {
          
        // Store the probabilities
        double[,] dp = new double[n + 1,6 * n + 1];
      
        // Precompute the probabilities
        // for values possible using 1 dice
        for(int i = 1; i <= 6; i++)
            dp[1,i] = 1 / 6.0;
      
        // Compute the probabilies
        // for all values from 2 to N
        for(int i = 2; i <= n; i++)
            for(int j = i - 1; j <= 6 * (i - 1); j++)
                for(int k = 1; k <= 6; k++)
                {
                    dp[i,j + k] += (dp[i - 1,j] *
                                     dp[1,k]);
                }
      
        // Print the result
        for(int i = n; i <= 6 * n; i++)
        {
            Console.WriteLine(i + " " +
                               Math.Round(dp[n,i] * 1000.0) /
                                                     1000.0);
        }
    }
 
  static void Main() {
    int n = 2;
  
    dicesSum(n);
  }
}
 
// This code is contributed by divyesh072019

Javascript

 
// JavaScript program to calculate
// the probability of all the
// possible values that can
// be obtained throwing N dices
 
function dicesSum(n)
{
      
    // Store the probabilities
    let dp = new Array(n+1);
    for (var i = 0; i < dp.length; i++) {
    dp[i] = new Array(2);
    }
    for (var i = 0; i < dp.length; i++) {
    for (var j = 0; j < 6 * n + 1; j++) {
        dp[i][j] = 0;
    }
    }
  
    // Precompute the probabilities
    // for values possible using 1 dice
    for(let i = 1; i <= 6; i++)
        dp[1][i] = 1 / 6.0;
  
    // Compute the probabilies
    // for all values from 2 to N
    for(let i = 2; i <= n; i++)
        for(let j = i - 1; j <= 6 * (i - 1); j++)
            for(let k = 1; k <= 6; k++)
            {
                dp[i][j + k] += (dp[i - 1][j] *
                                 dp[1][k]);
            }
  
    // Prlet the result
    for(let i = n; i <= 6 * n; i++)
    {
        document.write(i + " " +
                     Math.round(dp[n][i] * 1000.0) /
                                    1000.0 + "<br/>");
    }
}
  // Driver Code
     
    let n = 2;
      
    dicesSum(n);
                  

输出：

2 0.028
3 0.056
4 0.083
5 0.11
6 0.14
7 0.17
8 0.14
9 0.11
10 0.083
11 0.056
12 0.028

时间复杂度： O(N ² )
辅助空间： O(N ² )