Generate integer from 1 to 7 with equal probability

Given a function foo() that returns integers from 1 to 5 with equal probability, write a function that returns integers from 1 to 7 with equal probability using foo() only. Minimize the number of calls to foo() method. Also, use of any other library function is not allowed and no floating point arithmetic allowed.
Solution :
We know foo() returns integers from 1 to 5. How we can ensure that integers from 1 to 7 occur with equal probability? 
If we somehow generate integers from 1 to a-multiple-of-7 (like 7, 14, 21, …) with equal probability, we can use modulo division by 7 followed by adding 1 to get the numbers from 1 to 7 with equal probability.
We can generate from 1 to 21 with equal probability using the following expression.

 5*foo() + foo() -5 

Let us see how above expression can be used. 
1. For each value of first foo(), there can be 5 possible combinations for values of second foo(). So, there are total 25 combinations possible. 
2. The range of values returned by the above equation is 1 to 25, each integer occurring exactly once. 
3. If the value of the equation comes out to be less than 22, return modulo division by 7 followed by adding 1. Else, again call the method recursively. The probability of returning each integer thus becomes 1/7.
The below program shows that the expression returns each integer from 1 to 25 exactly once. 

C++

#include <stdio.h>

int main()
{
    int first, second;
    for ( first=1; first<=5; ++first )
        for ( second=1; second<=5; ++second )
            printf ("%d \n", 5*first + second - 5);
    return 0;
}

Java

// Java code to demonstrate
// expression returns each integer
// from 1 to 25 exactly once

class GFG {
    public static void main(String[] args)
    {
        int first, second;
        for ( first=1; first<=5; ++first )
            for ( second=1; second<=5; ++second )
            System.out.printf ("%d \n", 5*first + second - 5);
    }
}

// This code is contributed by
// Smitha Dinesh Semwal

Python3

# Python3 code to demonstrate
# expression returns each integer
# from 1 to 25 exactly once

if name == '__main__':

    for first in range(1, 6):
        for second in range(1, 6):
                print(5 * first + second - 5)

# This code is contributed by Smitha Dinesh Semwal.

C#

// C# code to demonstrate expression returns
// each integer from 1 to 25 exactly once
using System;

class GFG {

    public static void Main()
    {
        int first, second;

        for ( first = 1; first <= 5; ++first )
            for ( second = 1; second <= 5; ++second )
                Console.WriteLine ( 5*first + second - 5);
    }
}

// This code is contributed by Sam007.

PHP

<?php
// PHP program to Generate integer
// from 1 to 7 with equal probability

    $first;
    $second;
    for ( $first = 1; $first <= 5; ++$first )
        for ( $second = 1; $second <= 5; ++$second )
            echo 5 * $first + $second - 5, "\n";

// This code is contributed by ajit.
?>

Javascript

<script>

// Javascript Program to demonstrate
// expression returns each integer
// from 1 to 25 exactly once


// Driver Code

    let first, second;
    for ( first=1; first<=5; ++first )
        for ( second=1; second<=5; ++second ) {
             document.write( 5*first + second - 5);
             document.write("<br />");
        }

</script>

Output :

1
2
.
.
24
25

Time Complexity: O(1)

Auxiliary Space: O(1), since no extra space has been taken.

The below program depicts how we can use foo() to return 1 to 7 with equal probability. 

C++

// C++ program to Generate integer from
// 1 to 5 with equal probability
#include <stdio.h>

// given method that returns 1 to 5 with equal probability
int foo()
{
    // some code here
}

int my_rand() // returns 1 to 7 with equal probability
{
    int i;
    i = 5*foo() + foo() - 5;
    if (i < 22)
        return i%7 + 1;
    return my_rand();
}
// Driver code
int main()
{
    printf ("%d ", my_rand());
    return 0;
}

Java

// Java program to Generate integer from
// 1 to 5 with equal probability
class GfG
{

// given method that returns 1 to 5 with equal probability
static int foo()
{
    // some code here
    return 0;
}

// returns 1 to 7 with equal probability
public static int my_rand()
{
    int i;
    i = 5*foo() + foo() - 5;
    if (i < 22)
        return i%7 + 1;
    return my_rand();
}

// Driver code
public static void main (String[] args) {

    System.out.println(my_rand());
}
}

Python3

# Python3 program to Generate integer
# from 1 to 5 with equal probability
# given method that returns 1 to 5
# with equal probability
def foo():

    # some code here
    return 0;

# returns 1 to 7 with equal probability
def my_rand():
    i = 0;
    i = (5 * foo()) + (foo() - 5);
    if (i < 22):
        if(i < 0):
            return (i % 7 - 7) + 1;
        else:
            return (i % 7) + 1;

    return my_rand();

# Driver code
if __name__ == '__main__':
    print(my_rand());

# This code contributed by PrinciRaj1992

C#

// C# program to Generate integer from
// 1 to 5 with equal probability
using System;
class GfG
{

// given method that returns 1 to 5 with equal probability
static int foo()
{
    // some code here
    return 0;
}

// returns 1 to 7 with equal probability
public static int my_rand()
{
    int i;
    i = 5*foo() + foo() - 5;
    if (i < 22)
        return i%7 + 1;
    return my_rand();
}

// Driver code
public static void Main () {

    Console.Write(my_rand()+"\n");
}
}

PHP

<?php
// PHP program to Generate integer from
// 1 to 5 with equal probability
// given method that returns 1 to 5
// with equal probability
function foo()
{
    // some code here
}

// returns 1 to 7 with equal probability
function my_rand()
{
    $i;
    $i = 5 * foo() + foo() - 5;
    if ($i < 22)
        return $i % 7 + 1;
    return my_rand();
}

// Driver code
echo my_rand();

// This code is contributed by Tushil.
?>

Javascript

<script>

// Javascript program to Generate integer from
// 1 to 5 with equal probability

    // given method that returns 1 to 5 with equal probability
    function foo()
    {
        // some code here
        return 0;
    }

    // returns 1 to 7 with equal probability
    function my_rand()
    {
        let i;
        i = 5*foo() + foo() - 5;
        if (i < 22)
            return i%7 + 1;
        return my_rand();
    }

    // Driver code
    document.write(my_rand());

    // This code is contributed by rag2127

</script>

Output:

-4

Time Complexity: O(1)

Auxiliary Space: O(1), since no extra space has been taken.

This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above