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Generate 0 and 1 with 25% and 75% probability

Given a function rand50() that returns 0 or 1 with equal probability, write a function that returns 1 with 75% probability and 0 with 25% probability using rand50() only. Minimize the number of calls to rand50() method. Also, use of any other library function and floating point arithmetic are not allowed.



The idea is to use Bitwise OR. A bitwise OR takes two bits and returns 0 if both bits are 0, while otherwise the result is 1. So it has 75% probability that it will return 1.

Below is the implementation of above idea :

C++

// Program to print 1 with 75% probability and 0
// with 25% probability
#include <iostream>
using namespace std;
  
// Random Function to that returns 0 or 1 with
// equal probability
int rand50()
{
    // rand() function will generate odd or even
    // number with equal probability. If rand()
    // generates odd number, the function will
    // return 1 else it will return 0.
    return rand() & 1;
}
  
// Random Function to that returns 1 with 75%
// probability and 0 with 25% probability using
// Bitwise OR
bool rand75()
{
    return rand50() | rand50();
}
  
// Driver code to test above functions
int main()
{
    // Intialize random number generator
    srand(time(NULL));
  
    for(int i = 0; i < 50; i++)
        cout << rand75();
  
    return 0;
}

PHP

<?php
// Program to print 1 with 75% probability
// and 0 with 25% probability
  
// Random Function to that returns 0 or
// 1 with equal probability
function rand50()
{
      
    // rand() function will generate 
    // odd or even number with equal
    // probability. If rand() generates
    // odd number, the function will
    // return 1 else it will return 0.
    return rand() & 1;
}
  
// Random Function to that returns
// 1 with 75% probability and 0 
// with 25% probability using
// Bitwise OR
function rand75()
{
    return rand50() | rand50();
}
  
    // Driver Code
    // Intialize random 
    // number generator
    srand(time(NULL));
  
    for($i = 0; $i < 50; $i++)
        echo rand75();
  
// This code is contributed m_kit
?>

/div>

Output:

11101111110010010110011111111101111110111100011000

On similar lines, we can also use Bitwise AND. Since it returns 0 with 75% probability, we have to invert the result.

// Random Function to that returns 1 with 75% 
// probability and 0 with 25% probability using
// Bitwise AND
bool rand75() 
{
    return !(rand50() & rand50());
}

We can replace Bitwise OR and Bitwise AND operator by OR and AND operators as well –

// Random Function to that returns 1 with 75%
// probability and 0 with 25% probability using 
// OR or AND operator
int rand75()
{
    return !(rand50() && rand50());
    // return rand50() || rand50()
}

We can also achieve the result using left shift operator and Bitwise XOR

C++

// Program to print 1 with 75% probability and 0
// with 25% probability
#include <iostream>
using namespace std;
  
// Random Function to that returns 0 or 1 with
// equal probability
int rand50()
{
    // rand() function will generate odd or even
    // number with equal probability. If rand()
    // generates odd number, the function will
    // return 1 else it will return 0.
    return rand() & 1;
}
  
// Random Function to that returns 1 with 75%
// probability and 0 with 25% probability using
// left shift and Bitwise XOR
int rand75()
{
    // x is one of {0, 1}
    int x = rand50();
  
    x = x << 1;
  
    // x is now one of {00, 10}
  
    x = x ^ rand50();
  
    // x is now one of {00, 01, 10, 11}
  
    return (x > 0) ? 1 : 0;
}
  
// Driver code to test above functions
int main()
{
    // Intialize random number generator
    srand(time(NULL));
  
    for (int i = 0; i < 50; i++)
        cout << rand75();
  
    return 0;
}

PHP

<?php
// Program to print 1 with 
// 75% probability and 0
// with 25% probability
  
// Random Function to that 
// returns 0 or 1 with
// equal probability
function rand50()
{
    // rand() function will 
    // generate odd or even 
    // number with equal 
    // probability. If rand() 
    // generates odd number, 
    // the function will return
    // 1 else it will return 0.
    return rand() & 1;
}
  
// Random Function to that 
// returns 1 with 75% 
// probability and 0 with
// 25% probability using
// left shift and Bitwise XOR
function rand75()
{
    // x is one of {0, 1}
    $x = rand50();
  
    $x = $x << 1;
  
    // x is now one
    // of {00, 10}
  
    $x = $x ^ rand50();
  
    // x is now one of
    // {00, 01, 10, 11}
  
    return ($x > 0) ? 1 : 0;
}
  
// Driver code 
  
// Intialize random
// number generator
srand(time(NULL));
  
for ($i = 0; $i < 50; $i++)
    echo rand75();
      
// This code is contributed 
// by ajit
?>


Output:

01101110111011000111111111110001111011101110110110

Please note above solutions will produce different results every time we run them.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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