= Unchanging betting probability under sampling without replacement
{title2=$q(r,b)=r(r-1)/[(r+b)(r+b-1)]$}
In a uniformly shuffled deck with $r$ red and $b$ black cards remaining, the conditional probability that the next two are red is $q(r,b)$. If at least three cards remain, $q(r,b)$ is the weighted average of $q(r-1,b)$ and $q(r,b-1)$ after one reveal. Hence these probabilities form a bounded <martingale> until only two cards remain. Any adapted choice of exactly one betting time before that deadline has the same winning probability by bounded <optional stopping>. Waiting creates no advantage when a bet must eventually be made.
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