I have seen this question asked a lot but never seen a true concrete answer to it. So I am going to post one here which will hopefully help people understand why exactly there is "modulo bias" when using a random number generator, like rand() in C++.
So rand() is a pseudo-random number generator which chooses a natural number between 0 and RAND_MAX, which is a constant defined in cstdlib (see this article for a general overview on rand()).
Now what happens if you want to generate a random number between say 0 and 2. For the sake of explanation, lets say RAND_MAX was 10 and I decide that the best way to generate a random number between 0 and 2 is to do rand()%3. Assuming rand() does generate each number between 0 and 10 with equal probability, (this is arguable but for this post I will assume it does), why would rand()%3 not produce the numbers between 0 and 2 with equal probability? When rand() returns 0, 3, 6, or 9, rand()%3 == 0. When rand() returns 1, 4, 7, or 10, rand()%3 == 1. When rand() returns 2, 5, or 8, rand()%3 == 2. Now if we analyze this statistically, we very quickly see that the probability of getting a 0 is 4/11, 1 is 4/11 but 2 is 3/11. This does not generate the numbers between 0 and 2 with equal probability. Of course for small ranges this might not be the biggest issue but for a larger range this could skew the distribution, biasing the smaller numbers.
So when does rand()%n return a range of numbers from 0 to n-1 with equal probability? When RAND_MAX%n == n - 1. In this case, along with our earlier assumption rand() does return a number between 0 and RAND_MAX with equal probability, the modulo classes of n would also be equally distributed.
So how do we solve this problem? One way is to keep generating random numbers till you get a number in your desired range:
int x;
do
{
x = rand();
} while (x >= n);
Hope that helps everyone!
Works cited and further reading: