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 .

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 player A rolls the number p twice is . The likelihood that player A rolls p

and then a number less than p is ; the probability that this occurs is equal to the

probability that player A first rolls a number less than p before rolling p. Therefore the probability that a

number p is recognized as player A’s higher roll is .

Given that player A’s higher roll is p, player B must roll an integer q such that for

player A to win the game. As there are p-1 possibilities that lie within these bounds, the probability that

player B rolls q such that is . Therefore the probability that player A wins the game is

. 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 .

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 . Then she must

roll numbers less than p across her remaining n-c trials. Across all n trials, there are 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 different ways to do this for each p out of total possible

outcomes; therefore the probability that p is the highest number that player A rolls is

. Similarly, in the probability that an integer q between one and six is

recognized as player B’s highest roll is , where d is the number of times that

he rolls q.

Note that, in order for player A to win by rolling p, player B must roll an integer q across d trials

such that . Thus q can be expressed as p-f, where and f is an integer. The

probability that a specific p will allow player A to win is therefore

; thus the probability that

player A will win is ; note that this equation will only yield correct probabilities in a game consisting two

people playing with fair, six-sided dice.

To verify this equation, some game variations have been analyzed first by counting the number

of outcomes in which player A wins to determine the probability of player A winning in those scenarios.

Then, the probabilities of player A winning in each of the game types is calculated using the above

formula with the help of a custom-made script on Microsoft Excel:

Game Variation “Counted” Probability Calculated Probability (Equation)

A has one roll, B has one roll

A has two rolls, B has one roll

A has two rolls, B has two rolls

A has three rolls, B has one roll

Creating the Casino Games

In order to design a profitable casino game, this investigation will first analyze the optimal entry

fee made by the player and the optimal payouts by the casino in the simple game in which each player

rolls one die. In this adaptation, both the casino and player roll the die once; if the player rolls a number

that is greater than that of the casino, the player wins; however, if the player rolls a number less than or

equal to that of the casino, the player loses. Under this system, the casino has a probability of

winning and the player has a probability of winning. The ratio of the casino’s winning chances to the

player’s winning chances is ; therefore, if the fee to play is x and the payoff offered by the casino is

1.4x, the casino would just break even in the long term. Therefore, the payoff should be under 140% of

the fee to play so that the casino stands to profit. However, the payoff should also be of noticeably

greater value than the fee to play or else visitors will have no incentive to play the game. In this case,

the casino has two options. It may charge an exorbitant fee to play the game, in which case it can set the

payoff as being relatively close to the break-even payoff (e.g. by making the payoff 135% of the playing

fee), which would allow the casino to make a large profit even though the payoff is relatively high when

compared to the player’s payment. This would have the added advantage of inducing patrons to play

the game, as their expected earnings are quite high when compared to other games. However, the high

payment required to play the game may put some players off. Alternatively, the casino could make the

payoff close to the player’s payment (e.g. 110% of the entry fee) and have a fairly low playing fee. Thus

the casino would be expected to retain a greater percentage of player payments, and visitors may be

drawn in by the low payment to play. However, the low expected earnings value may again turn

potential players away. With respects to fairly rational-minded casino patrons, having a large entry fee

and higher expected earnings (or lower expected loss) would be a more attractive prospect in the long

run. Thus by setting the payoff at 130% to 135% of the payment to play, and by setting the payment at

an above-average value (e.g. $50), the casino can attract more players and make a larger profit.

This investigation will now consider a game where each player rolls their dice twice. If the

player’s largest roll is greater than the casino’s largest roll, then the player wins; in all other cases, the

casino wins. In the Investigating Dice Games section, the probability of the player winning was found to

be , and the probability of the casino winning is therefore . The ratio of the

probability of the casino winning to the probability of the player winning is , approximately 156%.

By the same logic as above, setting the player’s payment at around $50 and the payoff to be

approximately 150% of the player payment would incentivize many visitors to play and allow the casino

to reap a profit.

Finally, this investigation will consider a game with three players. In this game, each player puts

in an equal payment; 25% goes to the casino and 75% is to be split among the winners. To win, a player

must roll the highest number present; in case of a two-way tie, the two winners split the earnings

equally. If all three players roll the same number, all player payments are given to the casino. The

advantage of such a game is that it is fair; each player has an equal likelihood of winning, as each player

is rolling an identical die. Furthermore, the casino does not have to offer payouts to the players,

meaning that its expenses for running the game are minimized, but gains earnings no matter the results

of the game. Players will be incentivized to play the game because they still retain the majority of the

earnings should they win. Players may decide how much money to gamble, as in poker.

Summary

This investigation has analyzed various dice games, starting from the simple case where two

players A and B roll one die each, with A winning if their rolled number is larger than that of player B.

This game was further expanded to encompass the case in which the greater of two rolls made by player

A was counted against the roll made by player B, the case in which each player was able to roll the die

twice, and the case where the greatest of three rolls made by player A was counted against one roll

made by player B. A general formula for player A’s chances of winning were determined; not however

that this formula is limited to the instance in which all dice are fair and six sided, and each face on the

each dice is assigned a different number between one and six. Using this information, possible payouts

and player entry fees were modeled for some casino games built on the dice game template.

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.