Probability of getting all possible values on throwing N dices

Given an integer N denoting the number of dices, the task is to find the probability of every possible value that can be obtained by throwing N dices together.

Examples:

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]. 

Approach: The idea is to use Dynamic programming and DP table to store the probability of each possible value.  

• Store the probabilities of all the 6 numbers that can appear on throwing 1 dice.
• Now, for N=2, the probability for all possible sums between [2, 12] is equal to the sum of the product of the respective probability of the two numbers that add up to that sum. For example,

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) 

• Hence for N dices,

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) 

• Hence, in order to solve the problem, we need to fill dp[][] table from 2 to N using a top-down approach using the relation:

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

• Display all the probabilities stored for N as the answer.

Below is the implementation of the above approach:

C++

// C++ Program to calculate
// the probability of
// all the possible values
// that can be obtained
// throwing N dices

#include <bits/stdc++.h>
using namespace std;

void dicesSum(int n)
{
    // Store the probabilities
    vector<map<int, double> > 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 probabilities
    // 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 probabilities
    // 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):

    # Initialize 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 probabilities
    # 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 probabilities
        // 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

<script>

// 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 probabilities
    // 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]);
            }

    // Print 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);

</script>

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

Time Complexity: O(N²)
Auxiliary Space: O(N²)