Stop before a red card

Question

A standard deck (26 red, 26 black) is shuffled and dealt face up one card at a time. At any moment, before the next card is dealt, you may say "stop". You win if the next card is red. If you never say stop, the last card decides.

You can see every card that has been dealt, so it feels as though waiting for a moment when many reds remain should help.

Is there a strategy that wins with probability greater than one half? Prove your answer.

Accepted Answer

No. Every strategy, however clever, wins with probability exactly one half. The key observation. At any moment, the undealt cards are in a uniformly random order given what you have seen. So the next card and the bottom card of the deck are equally likely to be red, whatever the history: each is a uniformly random card from the remaining pile. This holds at every possible stopping moment. Replace "next" by "last". Consider any strategy. It decides when to stop based on the cards seen so far. Modify the game so that, when you stop, you win if the last card of the deck is red rather than the ne…

Stop before a red card

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