Perhaps I am missing something here but it seems quite intuitive to me that it has to be 50-50.<br>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][1] and a [martingale][2] <br>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.<br><br>Does this make sense?


  [1]: http://en.wikipedia.org/wiki/Markov_property
  [2]: http://en.wikipedia.org/wiki/Martingale_(probability_theory)