 Unchanging betting probability under sampling without replacement

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.