Unbiased Randomization with the Fisher-Yates Shuffle

In practice, say you're shuffling a 52-card deck - that's 52! orderings. Even if you have a 128-bit random seed, that's still nowhere near enough. You need at least a log2(52!) = 226 bit seed to cover all orderings, even assuming every seed produces a unique sequence of numbers, by the pigeonhole principle.

And it's even worse than that. Say you have a 226 bit seed. Well, you still won't have a uniform distribution. Why? Because 52! does not evenly divide 2^226. Even assuming a "perfect" PRNG (again: where every seed produces a unique sequence of numbers), you'll end up hitting some deck orderings less than others. In this case, it's a difference between some deck orderings being hit once and others being hit twice - which can make a big difference! You can mitigate this by either discarding and reinitializing your PRNG if the initial seed is >= 52!, or just choosing a bit length long enough that the effect becomes minimal.

It's like trying to pick a number between 0 and 45 with a 100 sided die - if you just take the number rolled mod 45, you'll end up with 0 through 9 picked more often than they should be. Instead, you have to roll until you get a number between 0 and 89, inclusive, then take mod 45.

(Oh, and as for the response of "just pick a larger seed", say you have a card game played with 4 decks (for example Shanghai rum with a bunch of players). Then you need a seed length of >=1307 bits. I cannot think of any PRNGs with a seed that large.)

"This is easy to verify for N = 2: You are just flipping a coin to decide if you swap 1 with 2. For N > 2, you just need to show that each of 1 through N has an equal chance of getting the k-th slot for 1 through N. In the first step, every number has a 1/N probability of getting into the first slot. For all other slots: the number has (1-1/N) chance of getting/staying there, then by induction, all slots are equally likely, hence (1-1/N)*1/(N-1) = 1/N. This completes the proof."

[1]: https://metacpan.org/pod/List::Util#values-shuffle-values [2]: http://www.perlmonks.org/?node_id=1869