Optimal Card Betting With a Known Deck
You have a standard deck of 52 cards (26 red, 26 black), shuffled randomly. Cards are revealed one at a time from the top. Before each reveal, you may bet any amount up to your current bankroll that the next card will be red. If you're right, you win that amount; if you're wrong, you lose it. You start with $\
$.
What strategy maximizes your expected final bankroll, and what is the expected value under optimal play?