Let’s play the following simple game: I will shuffle a standard 52-card deck, and deal one card at a time face up onto the table between us. You can say “Stop” at any time, at which point I will pay you an amount equal to the fraction of cards dealt so far that are red. For example, if you stop after seeing a single card, you win 1 if it is red, 0 if it is black, with expected return 1/2. Of course, you can wait until the entire deck is dealt, also with a guaranteed return of 1/2.
The problem is: can you do better than this? What is the optimal strategy for playing this game, and what is the corresponding expected return?
(This game is a variant of the following interesting game that I read about this week, which at first glance might seem even simpler, but for which optimal strategy remains an open problem: instead of dealing cards from a deck, I flip a coin repeatedly, and upon stopping I pay you an amount equal to the fraction of tosses that came up heads.)