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

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:

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:

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)