Well yeah, me neither: you know where you are with integers. :)
If it reaches the stage that the maths becomes important, you would certainly want to take a lot more care. One way to do that would be to avoid the simplistic "walk a list of (floating-point) weights" algorithm, and instead work to make each reciprocal exact: for each of the options in turn, give it an exact 1/votes chance to get chosen, and if it fails proceed to the next. On reaching the end of the list without a choice, start again (which doesn't necessarily have to be in the same order).
Getting an exact "1/votes" chance requires understanding a specific invariant of the RNG, the number of bits that it generates - exposed by perl via use Config; print $Config{randbits}. When something has 3 votes, for example, an RNG that generates 16 bits will slightly favour the value 0 (mod 3) by the ratio 21846:21845:21845. So for each option you would calculate $valid_lim = $votes * int(2**RNGbits / $votes) == 65535 in advance, and then re-roll any random value while ($rand = int(rand(2**$RNGbits))) >= $valid_lim before doing the modulus check choose_me() if ($rand % $votes) == 0, to remove that bias.
That means doing more work (and using up more of the available entropy), so you won't want to do this unless the vote requires this sort of critical accuracy.
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