If there is one card in this pile and no cards in the other, the probability of dropping the card from this pile is one. If instead there are some cards still in the other, a) the probability is less than one, and b) we move one step closer to the first state. So by construction it must be proportional - perhaps a poorly behaved proportionality, but that is still enough for the math to work.
reply> But not perfectly and continuously
replyIsn’t that where the randomness comes in?
The randomness comes from sampling the probabilities. The strange assumption is that the probabilities are exactly proportional to number of cards currently in each stack.
replyIt's the simplest model that gives the right result for simple cases (e.g. once one pile is empty, the remaining cards must come from the other pile; and when they're evenly split, it should be a coin flip). It also entails, for example, that when there are N cards in one pile and one in the other, the single card gets placed in each possible spot with equal probability. (This recalls the old trick for getting a random line from a text file without pre-counting anything.)
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