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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Introduction

        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 the numbers rolled by each player can be divided into three broad cases: either player A rolls the higher number, player B rolls the higher number, or players A and B roll the same number. Therefore, the probability that player A wins is the probability that the two players roll different numbers and that player A’s number is the higher of the two.

        As each die has faces numbered from one to six and there are two players present in the game, there are  possible outcomes for the game. Of these 36 outcomes, there are six ways for players A and B to roll the same number (they can both roll a 1, 2, 3, 4, 5, or 6). Therefore the probability that players A and B both roll the same number is , or .

        Because the probability that both players roll the same number is , it stands to reason that the probability in which both players roll different numbers is . Let the ordered pair (m,n) represent the case in which player A rolls a number m and player B rolls a number n such that m and n are integers, , and . Now consider the ordered pair (a,b) where . For every such pair (a,b), there exists another pair (b,a); thus there exists a one-to-one correspondence between the sets of (a,b) and (b,a). Therefore, the number of outcomes where player A rolls the higher number is equal to the number of outcomes in which player B rolls the higher number, or one-half of all the outcomes in which both players roll different numbers. Thus the probability that player A wins the game is .

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        Next, consider a variation of the above game in which player A may roll the die twice; if the higher of the two rolls is greater than player B’s roll, then player A wins. In order to determine the probability that player A wins, it is necessary to first find the probabilities of the outcomes of each of player A’s rolls.

        Now consider a specific whole number p between one and six inclusive. In order for p to be recognized as player A’s higher roll, it must be equal to or higher than player A’s other roll. The likelihood that ...

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