You can watch the deck go by and choose when to make your bet. What advantage does that freedom give you?
The Card You Haven’t Seen Yet
Riddle statement
A deck of 52 cards—26 red and 26 black—is shuffled uniformly at random: every ordering is equally likely.
Before any card is turned over, including the first, you may say “stop”. If you do not, the next card is revealed and discarded. When you say “stop”, the next card is revealed: you win if it is red and lose if it is black. You must decide before seeing it. If only one card remains, you must choose it.
You may use everything you have seen so far. Which strategy gives you the best chance of winning, and what is that chance?
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Solution
Every allowed strategy wins with probability 1/2. Stopping before the first card is already optimal, but so is every other strategy permitted by the rules.
We prove the stronger statement that with r red cards and b black cards remaining, any subsequent strategy wins with probability r/(r+b).
With one card, the claim is immediate. Suppose it holds for all smaller decks, and write N = r + b. Stopping now wins with probability r/N.
If we continue, a red card is discarded with probability r/N, leaving a success probability of (r−1)/(N−1). A black card is discarded with probability b/N, leaving r/(N−1). The induction assumption applies whatever we choose to do after seeing the discarded card. Thus continuing wins with probability
[r/N] × [(r−1)/(N−1)] + [b/N] × [r/(N−1)] = r/N.
This calculation covers states with both colours present. If only one colour remains, every choice has the same certain outcome. This completes the induction. Randomising between stopping and continuing cannot change their common value either, provided the random choice gives no information about unseen cards.
Initially r = b = 26, so the probability is 1/2.
After seeing many black cards, your conditional chance really can improve. But you cannot guarantee that you will reach such a favourable situation. Before play begins, the gains in some histories are balanced by the losses in others.