Perhaps I am missing something here but it seems quite intuitive to me that it has to be 50-50.
Think of it as coin tosses: As the resulting distribution is completely independent of the stopping time the proportion will in the limit always converge to the original distribution which is supposed to be 50-50. That is because the resulting stochastic process is markovian and a martingale
Or put another way: It doesn't matter when you stop tossing, the outcome will always add up to 50-50 because the coin doesn't have a memory.
Does this make sense?
vonjd
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