Flipping Cards Probability > You have a deck of cards, 26 red, 26 black. These are turned over, and at any point you may stop and exclaim "The next card is red.". If the next card is red you win £10. > > What's the optimal strategy? Prove this is the optimal strategy. I feel like the optimal strategy is whenever you are in a state that you flip more black than red cards you call out red card, but I do not know how to prove it, nor can I think of why it must be the absolute best strategy. Thanks for any help.

There is no optimal strategy (or rather, every strategy is optimal). Assuming your strategy involves exclaiming before the last card is drawn, your expected winnings are 5 pounds.

If you don't exclaim, your winnings are obviously zero.

Here's a reference: "Games People Don't Play" (Peter Winkler)