Why is rand()kiased?

Question

When reading how to use std::rand, I found this code on cppreference.com

int x = 7;
while(x > 6) 
    x = 1 + std::rand()/((RAND_MAX + 1u)/6);  // Note: 1         


        

Answer 1

One can think of a random number generator as working on a stream of binary digits. The generator turns the stream into numbers by slicing it up into chunks. If the std:rand function is working with a RAND_MAX of 32767, then it is using 15 bits in each slice.

When one takes the modules of a number between 0 and 32767 inclusive one finds that 5462 '0's and '1's but only 5461 '2's, '3's, '4's, and '5's. Hence the result is biased. The larger the RAND_MAX value is, the less bias there will be, but it is inescapable.

What is not biased is a number in the range [0..(2^n)-1]. You can generate a (theoretically) better number in the range 0..5 by extracting 3 bits, converting them to an integer in the range 0..7 and rejecting 6 and 7.

One hopes that every bit in the bit stream has an equal chance of being a '0' or a '1' irrespective of where it is in the stream or the values of other bits. This is exceptionally difficult in practice. The many different implementations of software PRNGs offer different compromises between speed and quality. A linear congruential generator such as std::rand offers fastest speed for lowest quality. A cryptographic generator offers highest quality for lowest speed.