Card Sharks

[Nate tha' Great]

The following is a problem based on the game show Card Sharks. It isn't a poker problem per se, but is reasonably interesting to think about, and probably has some implications in terms of bankroll management.

The contestant is dealt a row of five cards from a standard 52 card deck. The cards are not replaced. The contestant starts with a bankroll of $1,000.

The cards are revealed one at a time. The contestant must determine two things as each card is flipped up:

a) A guess as to whether the next card will be higher or lower. Aces are high, and a push is considered a loss.
b) The contestant may bet any amount from $0 to his current bankroll on his guess.

For example, a typical game might go like this.

The first card is a 4. The contestant bets $800 that the next card will be higher.

The next card is a 5, bringing the contestant's bankroll to $1,800. He bets $700 that the next card will be higher.

The third card is a Q, bringing the contestant's bankroll to $2,500. He bets $2,000 that the next card will be lower.

The fourth card is a J, bringing the bankroll to $4,500. The contestant bets $3,000 that the next card will be lower.

The fifth and final card is a J. The push is conisidered a loss and the contestant finishes the game with $1,500.

Here is the problem: suppose that the first card is a T. Obviously, the contestant should guess that the next card will be lower. But what should his wager be?

Assumption: the contestant is trying to maximize his final total and is neither risk-averse nor risk-loving.

p.s. Nope, haven't been playing too many real cards lately. I'll be back in the grind soon enough.