Find the expected value for a player of the following game who applies an expectation-maximizing strategy.
The player is given one dollar to start and may not bring additional money into to the game. A standard 52-card deck of cards is shuffled and dealt, and before each card is dealt, the player has the opportunity to bet, at even odds, on whether the card is red or black.
For example, if the player bets 50 cents that the first card off the top of the deck is red, and is correct, the player will have 1.50 available to wager on the next card. If the player subsequently bets 1.50 that the second card is black, and is wrong, the player then has no money, and cannot make bets for the rest of the game.
Find the expected value of the game for a player using an expectation-maximizing strategy. So this problem requires finding both an expectation maximizing strategy, as well as the expected value using that strategy.