Probability that the sum of all numbers obtained on throwing a dice N times lies between two given integers

19 June 2025

0

Given three integers N, A, and B, the task is to calculate the probability that the sum of numbers obtained on throwing the dice exactly N times lies between A and B.

Examples:

Input: N = 1, A = 2, B = 3
Output: 0.333333
Explanation: Ways to obtained the sum 2 by N ( = 1) throws of a dice is 1 {2}. Therefore, required probability = 1/6 = 0.33333

Input: N = 2, A = 3, B = 4
Output: 0.138889

Recursive Approach: Follow the steps below to solve the problem:

Calculate probabilities for all the numbers between A and B and add them to get the answer.
Call function find(N, sum) to calculate the probability for each number from a to b, where a number between a and b will be passed as sum.
- Base cases are:
  - If the sum is either greater than 6 * N or less than N, then return 0 as it’s impossible to have sum greater than N * 6 or less than N.
  - If N is equal to 1 and sum is in between 1 and 6, then return 1/6.
- Since at every state any number out of 1 to 6 in a single throw of dice may come, therefore recursion call should be made for the (sum up to that state – i) where 1≤ i ≤ 6.
- Return the resultant probability.

Recursion call:
$f(n, sum)=\sum_{i=1}^6\frac{f(n-1, sum-i)}{6}$

Below is the implementation of the above approach:

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the
// probability for the given
// sum to be equal to sum in
// N throws of dice
long double find(int N, int sum)
{
    // Base cases
    if (sum > 6 * N || sum < N)
        return 0;
 
    if (N == 1) {
 
        if (sum >= 1 && sum <= 6)
            return 1.0 / 6;
        else
            return 0;
    }
    long double s = 0;
    for (int i = 1; i <= 6; i++)
        s = s + find(N - 1, sum - i) / 6;
 
    return s;
}
 
// Driver Code
int main()
{
    int N = 4, a = 13, b = 17;
    long double probability = 0.0;
 
    for (int sum = a; sum <= b; sum++)
        probability = probability + find(N, sum);
 
    // Print the answer
    cout << fixed << setprecision(6) << probability;
    return 0;
}

Java

// Java program for the above approach 
import java.util.*;
class GFG
{
 
// Function to calculate the
// probability for the given
// sum to be equal to sum in
// N throws of dice
static double find(int N, int sum)
{
    // Base cases
    if (sum > 6 * N || sum < N)
        return 0;
    if (N == 1)
    {
        if (sum >= 1 && sum <= 6)
            return 1.0 / 6;
        else
            return 0;
    }
    double s = 0;
    for (int i = 1; i <= 6; i++)
        s = s + find(N - 1, sum - i) / 6;
    return s;
}
   
// Driver code
public static void main(String[] args)
{
    int N = 4, a = 13, b = 17;
    double probability = 0.0;
    for (int sum = a; sum <= b; sum++)
        probability = probability + find(N, sum);
 
    // Print the answer
    System.out.format("%.6f", probability);
}
}
 
// This code is contributed by code_hunt.

Python3

# Python 2 program for above approach
 
# Function to calculate the
# probability for the given
# sum to be equal to sum in
# N throws of dice
def find(N, sum):
 
    # Base cases
    if (sum > 6 * N or sum < N):
        return 0
    if (N == 1):
        if (sum >= 1 and sum <= 6):
            return 1.0 / 6
        else:
            return 0
    s = 0
    for i in range(1, 7):
        s = s + find(N - 1, sum - i) / 6
    return s
 
# Driver Code
if __name__ == "__main__":
    N = 4
    a = 13
    b = 17
    probability = 0.0
    for sum in range(a, b + 1):
        probability = probability + find(N, sum)
 
    # Print the answer
    print(round(probability, 6))
 
    # This code is contributed by chitranayal.

C#

// C# program for the above approach 
using System;
class GFG 
{
     
    // Function to calculate the
    // probability for the given
    // sum to be equal to sum in
    // N throws of dice
    static double find(int N, int sum)
    {
       
        // Base cases
        if (sum > 6 * N || sum < N)
            return 0;
        if (N == 1)
        {
            if (sum >= 1 && sum <= 6)
                return 1.0 / 6;
            else
                return 0;
        }
        double s = 0;
        for (int i = 1; i <= 6; i++)
            s = s + find(N - 1, sum - i) / 6;
        return s;
    }
 
  // Driver code
  static void Main()
  {
    int N = 4, a = 13, b = 17;
    double probability = 0.0;
    for (int sum = a; sum <= b; sum++)
        probability = probability + find(N, sum);
  
    // Print the answer
    Console.WriteLine(Math.Round(probability,6));
  }
}
 
// This code is contributed by divyeshrabadiya07

Javascript

<script>
 
    // Javascript program for the above approach
     
    // Function to calculate the
    // probability for the given
    // sum to be equal to sum in
    // N throws of dice
    function find(N, sum)
    {
        
        // Base cases
        if (sum > 6 * N || sum < N)
            return 0;
        if (N == 1)
        {
            if (sum >= 1 && sum <= 6)
                return 1.0 / 6;
            else
                return 0;
        }
        let s = 0;
        for (let i = 1; i <= 6; i++)
            s = s + find(N - 1, sum - i) / 6;
        return s;
    }
     
    let N = 4, a = 13, b = 17;
    let probability = 0.0;
    for (let sum = a; sum <= b; sum++)
        probability = probability + find(N, sum);
   
    // Print the answer
    document.write(probability.toFixed(6));
   
</script>

Output:

0.505401

Time Complexity: O((b-a+1)6ⁿ)
Auxiliary Space: O(1)

Dynamic Programming Approach: The above recursive approach needs to be optimized by dealing with the following overlapping subproblems and optimal substructure:

Overlapping Subproblems:
Partial recursion tree for N=4 and sum=15:

Optimal Substructure:
For every state, recur for other 6 states, so the recursive definition of f(N, sum) is:

$dp[N][sum]=\sum_{i=1}^6\frac{dp[N-1][sum-i]}{6}$

Top-Down Approach:

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;
float dp[105][605];
 
// Function to calculate the
// probability for the given
// sum to be equal to sum in
// N throws of dice
float find(int N, int sum)
{
    if (dp[N][sum])
        return dp[N][sum];
 
    // Base cases
    if (sum > 6 * N || sum < N)
        return 0;
    if (N == 1) {
        if (sum >= 1 && sum <= 6)
            return 1.0 / 6;
        else
            return 0;
    }
    for (int i = 1; i <= 6; i++)
        dp[N][sum] = dp[N][sum]
                     + find(N - 1, sum - i) / 6;
    return dp[N][sum];
}
 
// Driver Code
int main()
{
    int N = 4, a = 13, b = 17;
    float probability = 0.0;
 
    // Calculate probability of all
    // sums from a to b
    for (int sum = a; sum <= b; sum++)
        probability = probability + find(N, sum);
 
    // Print the answer
    cout << fixed << setprecision(6) << probability;
    return 0;
}

Java

// Java program for above approach
class GFG 
{
    static float[][] dp = new float[105][605];
 
    // Function to calculate the
    // probability for the given
    // sum to be equal to sum in
    // N throws of dice
    static float find(int N, int sum) 
    {
        if (N < 0 | sum < 0)
            return 0;
        if (dp[N][sum] > 0)
            return dp[N][sum];
 
        // Base cases
        if (sum > 6 * N || sum < N)
            return 0;
        if (N == 1) {
            if (sum >= 1 && sum <= 6)
                return (float) (1.0 / 6);
            else
                return 0;
        }
        for (int i = 1; i <= 6; i++)
            dp[N][sum] = dp[N][sum] + find(N - 1, sum - i) / 6;
        return dp[N][sum];
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int N = 4, a = 13, b = 17;
        float probability = 0.0f;
 
        // Calculate probability of all
        // sums from a to b
        for (int sum = a; sum <= b; sum++)
            probability = probability + find(N, sum);
 
        // Print the answer
        System.out.printf("%.6f", probability);
    }
}
 
// This code is contributed by shikhasingrajput

Python3

# Python program for above approach
dp = [[0 for i in range(605)] for j in range(105)];
 
# Function to calculate the
# probability for the given
# sum to be equal to sum in
# N throws of dice
def find(N, sum):
    if (N < 0 | sum < 0):
        return 0;
    if (dp[N][sum] > 0):
        return dp[N][sum];
 
    # Base cases
    if (sum > 6 * N or sum < N):
        return 0;
    if (N == 1):
        if (sum >= 1 and sum <= 6):
            return (float)(1.0 / 6);
        else:
            return 0;
 
    for i in range(1,7):
        dp[N][sum] = dp[N][sum] + find(N - 1, sum - i) / 6;
    return dp[N][sum];
 
# Driver Code
if __name__ == '__main__':
    N = 4; a = 13; b = 17;
    probability = 0.0
    f = 0;
 
    # Calculate probability of all
    # sums from a to b
    for sum in range(a,b+1):
        probability = probability + find(N, sum);
 
    # Print the answer
    print("%.6f"% probability);
 
# This code is contributed by 29AjayKumar

C#

// C# program for above approach
using System;
using System.Collections.Generic;
public class GFG 
{
  static float[,] dp = new float[105, 605];
 
  // Function to calculate the
  // probability for the given
  // sum to be equal to sum in
  // N throws of dice
  static float find(int N, int sum) 
  {
    if (N < 0 | sum < 0)
      return 0;
    if (dp[N, sum] > 0)
      return dp[N, sum];
 
    // Base cases
    if (sum > 6 * N || sum < N)
      return 0;
    if (N == 1) {
      if (sum >= 1 && sum <= 6)
        return (float) (1.0 / 6);
      else
        return 0;
    }
    for (int i = 1; i <= 6; i++)
      dp[N, sum] = dp[N, sum] + find(N - 1, sum - i) / 6;
    return dp[N, sum];
  }
 
  // Driver Code
  public static void Main(String[] args)
  {
    int N = 4, a = 13, b = 17;
    float probability = 0.0f;
 
    // Calculate probability of all
    // sums from a to b
    for (int sum = a; sum <= b; sum++)
      probability = probability + find(N, sum);
 
    // Print the answer
    Console.Write("{0:F6}", probability);
  }
}
 
// This code is contributed by 29AjayKumar

Javascript

<script>
 
// Javascript program for above approach
 
var dp = Array(105).fill().map(()=>Array(605).fill(0.0));
 
// Function to calculate the
// probability for the given
// sum to be equal to sum in
// N throws of dice
function find(N, sum)
{
    if (N < 0 | sum < 0)
        return 0;
    if (dp[N][sum] > 0)
        return dp[N][sum];
 
    // Base cases
    if (sum > 6 * N || sum < N)
        return 0;
    if (N == 1)
    {
        if (sum >= 1 && sum <= 6)
            return  (1.0 / 6);
        else
            return 0;
    }
     
    for(var i = 1; i <= 6; i++)
        dp[N][sum] = dp[N][sum] + 
           find(N - 1, sum - i) / 6;
            
    return dp[N][sum];
}
 
// Driver Code
var N = 4, a = 13, b = 17;
var probability = 0.0;
 
// Calculate probability of all
// sums from a to b
for(sum = a; sum <= b; sum++)
    probability = probability + find(N, sum);
 
// Print the answer
document.write(probability.toFixed(6));
 
// This code is contributed by umadevi9616
 
</script>

Output:

0.505401

Time Complexity: O(n*sum)
Auxiliary Space: O(n*sum)

Bottom-Up Approach:

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;
float dp[105][605];
 
// Function to calculate probability
// that the sum of numbers on N throws
// of dice lies between A and B
float find(int N, int a, int b)
{
    float probability = 0.0;
 
    // Base case
    for (int i = 1; i <= 6; i++)
        dp[1][i] = 1.0 / 6;
 
    for (int i = 2; i <= N; i++) {
 
        for (int j = i; j <= 6 * i; j++) {
 
            for (int k = 1; k <= 6; k++) {
 
                dp[i][j] = dp[i][j]
                           + dp[i - 1][j - k] / 6;
            }
        }
    }
 
    // Add the probability for all
    // the numbers between a and b
    for (int sum = a; sum <= b; sum++)
        probability = probability + dp[N][sum];
 
    return probability;
}
 
// Driver Code
int main()
{
    int N = 4, a = 13, b = 17;
 
    float probability = find(N, a, b);
 
    // Print the answer
    cout << fixed << setprecision(6) << probability;
    return 0;
}

Java

// Java program for above approach
import java.util.*;
 
class GFG{
static float [][]dp = new float[105][605];
 
// Function to calculate probability
// that the sum of numbers on N throws
// of dice lies between A and B
static float find(int N, int a, int b)
{
    float probability = 0.0f;
 
    // Base case
    for (int i = 1; i <= 6; i++)
        dp[1][i] = (float) (1.0 / 6);
    for (int i = 2; i <= N; i++) 
    {
        for (int j = i; j <= 6 * i; j++)
        {
            for (int k = 1; k <= 6 && k <= j; k++)
            {
                dp[i][j] = dp[i][j]
                           + dp[i - 1][j - k] / 6;
            }
        }
    }
 
    // Add the probability for all
    // the numbers between a and b
    for (int sum = a; sum <= b; sum++)
        probability = probability + dp[N][sum];
    return probability;
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 4, a = 13, b = 17;
    float probability = find(N, a, b);
 
    // Print the answer
    System.out.printf("%.6f",probability);
}
}
 
// This codeis contributed by shikhasingrajput

Python3

# Python3 program for above approach
 
dp = [[0 for i in range(605)] for j in range(105)] 
 
# Function to calculate probability
# that the sum of numbers on N throws
# of dice lies between A and B
def find(N, a, b) :
 
    probability = 0.0
  
    # Base case
    for i in range(1, 7) : 
        dp[1][i] = 1.0 / 6
  
    for i in range(2, N + 1) :
  
        for j in range(i, (6*i) + 1) :
  
            for k in range(1, 7) :
  
                dp[i][j] = dp[i][j] + dp[i - 1][j - k] / 6
  
    # Add the probability for all
    # the numbers between a and b
    for Sum in range(a, b + 1) :
        probability = probability + dp[N][Sum]
  
    return probability
     
N, a, b = 4, 13, 17
 
probability = find(N, a, b)
 
# Print the answer
print('%.6f'%probability) 
 
# This code is contributed by divyesh072019.

C#

// C# program for above approach
using System;
public class GFG
{
static float [,]dp = new float[105, 605];
 
// Function to calculate probability
// that the sum of numbers on N throws
// of dice lies between A and B
static float find(int N, int a, int b)
{
    float probability = 0.0f;
 
    // Base case
    for (int i = 1; i <= 6; i++)
        dp[1, i] = (float) (1.0 / 6);
    for (int i = 2; i <= N; i++) 
    {
        for (int j = i; j <= 6 * i; j++)
        {
            for (int k = 1; k <= 6 && k <= j; k++)
            {
                dp[i, j] = dp[i, j]
                           + dp[i - 1, j - k] / 6;
            }
        }
    }
 
    // Add the probability for all
    // the numbers between a and b
    for (int sum = a; sum <= b; sum++)
        probability = probability + dp[N, sum];
    return probability;
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 4, a = 13, b = 17;
    float probability = find(N, a, b);
 
    // Print the answer
    Console.Write("{0:F6}",probability);
}
}
 
// This code is contributed by shikhasingrajput

Javascript

<script>
 
// Javascript program of the above approach
let dp = new Array(105);
 
// Loop to create 2D array using 1D array
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 < dp.length; j++) 
    {
        dp[i][j] = 0;
    }
}
  
// Function to calculate probability
// that the sum of numbers on N throws
// of dice lies between A and B
function find(N, a, b)
{
    let probability = 0.0;
  
    // Base case
    for(let i = 1; i <= 6; i++)
        dp[1][i] =  (1.0 / 6);
         
    for(let i = 2; i <= N; i++)
    {
        for(let j = i; j <= 6 * i; j++)
        {
            for(let k = 1; k <= 6 && k <= j; k++)
            {
                dp[i][j] = dp[i][j] + 
                           dp[i - 1][j - k] / 6;
            }
        }
    }
  
    // Add the probability for all
    // the numbers between a and b
    for(let sum = a; sum <= b; sum++)
        probability = probability + dp[N][sum];
         
    return probability;
}
 
// Driver Code
let N = 4, a = 13, b = 17;
let probability = find(N, a, b);
 
// Print the answer
document.write(probability);
 
// This code is contributed by chinmoy1997pal
 
</script>

Output:

0.505401

Time Complexity: O(N * sum)
Auxiliary Space: O(N * sum)

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