Probability of obtaining Prime Numbers as product of values obtained by throwing N dices

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Given an integer N denoting the number of dices, the task is to find the probability of the product of numbers appearing on the top faces of N thrown dices being a prime number. All N dices must be thrown simultaneously.

Examples:

Input: N = 2
Output: 6 / 36
Explanation:
On throwing N(=2) dices simultaneously, the possible outcomes on the top faces of N(=2) dices having product equal to a prime number are: {(1, 2), (1, 3), (1, 5), (2, 1), (3, 1), (5, 1)}.
Therefore, the count of favourable outcomes = 6 and the count of the sample space is = 36
Therefore, the required output is (6 / 36)

Input: N = 3
Output: 9 / 216

Naive Approach: The simplest approach to solve this problem is to generate all possible outcomes on the top faces of N dices by throwing N dices simultaneously and for each possible outcome check if the product of numbers on the top faces is a prime number or not. If found to be true then increment the counter. Finally, print the probability of getting the product of numbers on the top faces as a prime number.

Time Complexity: O(6^N * N)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach the idea is to use the fact that the product of N number is a prime number only if (N – 1) numbers are 1 and a remaining number is a prime number. Following are the observations:

If the product of N numbers is a prime number then the value of (N – 1) numbers must be 1 and the remaining number must be a prime number.
Total count of prime numbers in the range [1, 6] is 3.
Therefore, the total number of outcomes in which the product of N numbers on the top faces as a prime number = 3 * N.
P(E) = N(E) / N(S)
P(E) = probability of getting the product of numbers on the top faces of N dices as a prime number.
N(E) = total count of favourable outcomes = 3 * N
N(S) = total number of events in the sample space = 6^N

Follow the steps below to solve this problem:

Initialize a variable, say N_E to store the count of favorable outcomes.
Initialize a variable, say N_S to store the count of sample space.
Update N_E = 3 * N.
Update N_S = 6^N.
Finally, print the value of (N_E / N_S).

Below is the implementation of the above approach

C++

// C++ program to implement
// the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
 
// Function to find the value 
// of power(X, N)
long long int power(long long int x,
                    long long int N)
{
    // Stores the value
    // of (X ^ N)
    long long int res = 1;
 
    // Calculate the value of 
    // power(x, N)
    while (N > 0) {
         
       // If N is odd
       if(N & 1) {
            
           //Update res
           res = (res * x);
       }
        
       //Update x
       x = (x * x);
        
       //Update N
       N = N >> 1;
        
    }
    return res;
}
 
// Function to find the probability of
// obtaining a prime number as the
// product of N thrown dices 
void probablityPrimeprod(long long int N) 
{
    // Stores count of favorable outcomes
    long long int N_E = 3 * N;
     
    // Stores count of sample space
    long long int N_S = power(6, N);
     
    // Print the required probability
    cout<<N_E<<" / "<<N_S;
}
 
// Driver code
int main()
{
    long long int N = 2;
    probablityPrimeprod(N);
}

Java

// Java program to implement
// the above approach
import java.util.*;
 
class GFG{
     
// Function to find the value 
// of power(X, N)
static int power(int x, int N)
{
     
    // Stores the value
    // of (X ^ N)
    int res = 1;
 
    // Calculate the value of 
    // power(x, N)
    while (N > 0)
    {
         
        // If N is odd
        if (N % 2 == 1) 
        {
             
            // Update res
            res = (res * x);
        }
         
        // Update x
        x = (x * x);
         
        // Update N
        N = N >> 1;
    }
    return res;
}
 
// Function to find the probability of
// obtaining a prime number as the
// product of N thrown dices 
static void probablityPrimeprod(int N) 
{
     
    // Stores count of favorable outcomes
    int N_E = 3 * N;
     
    // Stores count of sample space
    int N_S = power(6, N);
     
    // Print the required probability
    System.out.print(N_E + " / " + N_S);
}
 
// Driver code
public static void main(String[] args)
{
    int N = 2;
     
    probablityPrimeprod(N);
}
}
 
// This code is contributed by Amit Katiyar

Python3

# Python3 program to implement
# the above approach
 
# Function to find the value
# of power(X, N)
def power(x, N):
     
    # Stores the value
    # of (X ^ N)
    res = 1
 
    # Calculate the value of
    # power(x, N)
    while (N > 0):
 
        # If N is odd
        if (N % 2 == 1):
             
            # Update res
            res = (res * x)
 
        # Update x
        x = (x * x)
 
        # Update N
        N = N >> 1
 
    return res
 
# Function to find the probability of
# obtaining a prime number as the
# product of N thrown dices
def probablityPrimeprod(N):
     
    # Stores count of favorable outcomes
    N_E = 3 * N
 
    # Stores count of sample space
    N_S = power(6, N)
 
    # Print required probability
    print(N_E, " / ", N_S)
 
# Driver code
if __name__ == '__main__':
     
    N = 2
 
    probablityPrimeprod(N)
 
# This code is contributed by 29AjayKumar

C#

// C# program to implement
// the above approach
using System;
class GFG{
     
// Function to find the 
// value of power(X, N)
static int power(int x, 
                 int N)
{    
  // Stores the value
  // of (X ^ N)
  int res = 1;
 
  // Calculate the value
  // of power(x, N)
  while (N > 0)
  {
    // If N is odd
    if (N % 2 == 1) 
    {
      // Update res
      res = (res * x);
    }
 
    // Update x
    x = (x * x);
 
    // Update N
    N = N >> 1;
  }
  return res;
}
 
// Function to find the probability 
// of obtaining a prime number as 
// the product of N thrown dices 
static void probablityPrimeprod(int N) 
{    
  // Stores count of favorable 
  // outcomes
  int N_E = 3 * N;
 
  // Stores count of sample 
  // space
  int N_S = power(6, N);
 
  // Print the required 
  // probability
  Console.Write(N_E + " / " + N_S);
}
 
// Driver code
public static void Main(String[] args)
{
  int N = 2;
  probablityPrimeprod(N);
}
}
 
// This code is contributed by Princi Singh

Javascript

<script>
 
// JavaScript program to implement the above approach
 
// Function to find the value 
// of power(X, N)
function power(x, N)
{
     
    // Stores the value
    // of (X ^ N)
    let res = 1;
 
    // Calculate the value of 
    // power(x, N)
    while (N > 0)
    {
         
        // If N is odd
        if (N % 2 == 1) 
        {
             
            // Update res
            res = (res * x);
        }
         
        // Update x
        x = (x * x);
         
        // Update N
        N = N >> 1;
    }
    return res;
}
 
// Function to find the probability of
// obtaining a prime number as the
// product of N thrown dices 
function probablityPrimeprod(N) 
{
     
    // Stores count of favorable outcomes
    let N_E = 3 * N;
     
    // Stores count of sample space
    let N_S = power(6, N);
     
    // Print the required probability
    document.write(N_E + " / " + N_S);
}
 
// Driver Code 
    let N = 2;
    probablityPrimeprod(N);
     
    // This code is contributed by susmitakunndugoaldanga.
 
</script>

Output:

6 / 36

Time Complexity: O(log₂N)
Auxiliary Space: O( 1 )

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