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

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(6N * 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 = 6N

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 = 6N.
• 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 ` `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 probablity` `    ``cout<

## 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 probablity` `    ``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)`   `    ``# Prthe required probablity` `    ``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 ` `  ``// probablity` `  ``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`

Output:

```6 / 36

```

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

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