Biased random number generation

Melissa O'Neill has a new post on generating random numbers from a given range. She gives the example of wanting to pick a card from a deck of 52 by first generating a 32-bit random integer, then taking the remainder when dividing by 52. There's a slight bias because 2³² is not a multiple of 52.

Since 2³² = 82595524*52 + 48, there are 82595525 ways to generate the numbers 0 through 47, but only 82595524 ways to generate the numbers 48 through 51. As Melissa points out in her post, the bias here is small, but the bias increases linearly with the size of our "deck." To clarify, it is the relative bias that increases, not the absolute bias.

Suppose you want to generate a number between 0 and M, where M is less than 2³² and not a power of 2. There will be 1 + a2³²/Ma ways to generate a 0, but a2³²/Ma ways to generate M-1. The difference in the probability of generating 0 vs generating M-1 is 1/2³², independent of M. However, the ratio of the two probabilities is 1 + 1/a2³²/Ma or approximately 1 + M/2³².

As M increases, both the favored and unfavored outcomes become increasingly rare, but ratio of their respective probabilities approaches 2.

Whether this makes any practical difference depends on your context. In general, the need for random number generator quality increases with the volume of random numbers needed.

Under conventional assumptions, the sample size necessary to detect a difference between two very small probabilities p₁ and p₂ is approximately

8(p₁ + p₂)/(p₁ - p₂)^2

and so in the card example, we would have to deal roughly 6 i- 10¹⁸ cards to detect the bias between one of the more likely cards and one of the less likely cards.

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