Actually, selecting with an exact probability of 1/3 (or any other rational fraction less than one) is feasible with even a source of single random bits. I saw the technique described once in "Mathematical Games" in Scientific American once.

Represent 1/3 as a binary fraction: .01010101... and generate random numbers bit by bit, starting from the decimal point, calling this a new binary fraction. If the next bit we generate is 1 where there is a 0 in the target, then quit -- we are over 1/3. If the next bit is 0 where there is a 1 in the target, then quit -- we are under 1/3 and we can accept the case. If it's equal, keep generating bits. The probability we will have to continue adding bits halves with each bit.

This approach can make the Fisher-Yates shuffle arbitrarily accurate. It would be possible, but messy, to apply it to sort values that compared equal, too. With this enhancement both shuffles should be fair, but Fisher-Yates wins by being O(N) instead of O(NlogN).

In reply to Re^6: Functional shuffle by tall_man
in thread Functional shuffle by Roy Johnson

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