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The purpose of this investigation is to create and model a dice-based casino game using probability. In order to be successful, this game must be able to allow the casino to profit from running it,

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        The purpose of this investigation is to create and model a dice-based casino game using probability. In order to be successful, this game must be able to allow the casino to profit from running it, but must also be attractive to potential players. In order to build such a game, this investigation will first analyze a game played between two people, each of whom rolls a single die. Next, this simple game will be expanded to consider games with more than two players and games in which some or all players can roll their die multiple times. Finally, a casino game based on these investigations will be created where probability is considered to determine the entry fee and payout of a game.

        This investigation assumes that all dice used by players are fair, six-sided dice.

Investigating Dice Games

        In order to begin this investigation, first consider a simple game played between two players, A and B. Each player may roll a die once, and player A wins if their number is higher than that of player B. The outcomes of

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 such that image05.pngis image06.png. Therefore the probability that player A wins the game is image07.png. Using a homemade program on Microsoft Excel, the probability that player A wins the game (that is, the highest number that she has rolled exceeds the highest number that player B has rolled) was calculated to be image08.png.

        Now consider the game where player A may roll her die n times and player B may roll his die m times, with n and m being positive integers that are not necessarily distinct. In order for a specific number p to be recognized as player A’s highest roll, it must be equal to or greater than all of the other rolls that player A makes. There are n ways that this could happen: player A may roll p anywhere from one to n times. Suppose that player A rolls p across c trials, where image09.png. Then she must roll numbers less than p across her remaining n-c trials. Across all n trials, there are image10.png ways to arrange the p’s that player A rolls. Each of the remaining slots may be filled with any integer between one and p; therefore there are image12.png different ways to do this for each p

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        This investigation has made use of a custom-made program on Microsoft Excel to calculate probabilities using the general formula. Another program was used to count the probability of player A winning in the variations of the dice game in which both players roll twice and in which player A rolls three times but player B rolls once.

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