Objective
The objective of this project is to prove the best strategy for playing Hi-Lo using basic probability. Using computer simulations, you can verify that a particular strategy is correct and show what happens to the odds of winning when "counting cards."
Introduction
Hi-Lo is a very simple card game. A dealer ("the house") starts with a deck of cards and turns over the top card. The player then guesses whether the next card in the deck will be higher or lower than that card. This process of turning over a card and guessing high or low continues through the rest of the deck. The best strategy allows the player to guess correctly more than 70% of the time. Interestingly, a consequence of the Law of Large Numbers is that remembering which cards have come up already - ie "counting cards" - does not substantially increase a player's chances of guessing correctly. This game becomes more complex if the player can bet on his or her guesses. The player can use a simple strategy to increase his or her expected winnings by 50% compared to always betting the same amount. More significantly, the player can also "count cards" and pick up an even larger advantage.
Terms, Concepts and Questions to Start Background Research
Probability theory, card counting, the law of large numbers
Experimental Procedure
First, you should play the game a bit by yourself. Develop your own strategy for playing the game (for example, if the card is greater than 7 you always guess lower, if the card is less than 7 you always guess higher) and then test it out by taking a deck of cards and keeping track of how often you can correctly guess whether the next card is higher or lower than the one you turned over. If you can guess right more than 70% of the time, you've probably got the right strategy. The next thing you need to do is pick a programming language. If you've never programmed before, you should start with QBASIC, which is available for free at many internet sites, and is as close to English as any programming language. You'll write a computer simulation that will play thousands of hands of this card game Your simulation program needs two parts - the first is a shuffling routine to make sure the deck is random. The second is the actual game-playing strategy. You'll also have to write some data collection routines so you know how many times you've won or lost, and on which cards. Programming the strategy is the most involved part of this project, and can lead to a lot of results about how to play the game.
Variations
A more advanced project would examine different betting and card counting strategies to determine the optimal betting strategy in different circumstances. You can also determine how much "the house" should pay to a player who guesses correctly, how many decks "the house" should use to discourage card counting, and how far the dealer should deal into the decks before shuffling and starting over.
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