Mathematics of dice games

Summary: Probability is the key factor for winning any dice game.

Dice games use one or more dice as central components of the activity, which excludes board games using dice solely as random devices to determine moves. The definition can be murky, as in the case of Backgammon, dice outcomes determine a player’s moves and are integral parts of game strategies. Historically, dice games involving gambling led to the creation of probability.

History

Archaeological evidence from as early as 6000 b.c.e. shows that dice games were part of early cultures, where dice were cast to invoke personal divinations. The notion of “luck” was not involved, with the dice rolls controlled by the gods. Gamblers still refer to Fortuna, the Roman goddess and Jupiter’s daughter, as their “Lady Luck.”

The ancient die differed from the six-sided cube bearing pips, as the number of sides varied with the materials used, including fruit pits, nut shells, pebbles, and animal knucklebones. The latter, with four sides involving different probabilities, led to the phrase “rolling the bones.”

Compulsive gambling and dice games have always been connected, being traced to Egyptian pharaohs, Chinese leaders, Roman emperors, Greek elite, European academics, and English kings. On the request of professional gamblers in the fifteenth and sixteenth centuries, mathematicians such as Fra Luca Bartolomeo de Pacioli and Girolamo Cardano began to study the probabilities of winning dice games. In the seventeenth century, correspondence between Blaise Pascal and Pierre Fermat ultimately solved the “problem of points” and established basic principles of probability.

The problem of points involves a dice game between two players; multiple rounds are played with each player having an equal chance of winning on each roll. If the game was interrupted before either player had won the necessary number of rounds, gamblers could not determine the “fair” division of stakes based on current scores. Fermat and Pascal’s solution analyzed the probability of dice rolls and each player winning the pot.

Types of Dice Games

The simplest dice game involves a single die, where the winner is the person rolling the highest number. This can be extended to rolls of multiple dice, with the player’s score being the sum or product of the numbers shown. Since these dice games involve only luck, gamblers prefer variations with elements of strategy.

The dice game craps involves strategy, as the “shooter” controls the number of dice rolls and betting options. Though craps is complex, key elements can be explained. Mathematically, each roll of two dice has 36 possible outcomes with shown totals ranging from “2” to “12”. However, the probabilities of the totals vary, as the probability of a “2” (known as “snake eyes”) or “12” (known as “boxcars”) is 1/36, while the probability of a “7” is 6/36. Prior to the first “come out roll,” players bet on the “Pass Line” or “Don’t Pass Line.” If the “shooter” then rolls a “7” or “11,” the “Pass Line” bet wins double their amount and the “Don’t Pass Line” bet is lost. However, if the initial roll is a “2,” “3,” or “12,” the “Pass Line” bet is lost, while the “Don’t Pass Line” bet is doubled if a “2” or “3” shows and is returned if a “12” (“push”) shows. A sum of “4,” “5,” “6,” “8,” “9,” or “10” becomes the “point” number, which the shooter tries to duplicate on the second roll. If the point number is made, the point bet is won and additional rolls can be made. But, if a “7” is rolled before the point number, the shooter “craps out” and a different shooter starts a new round. Craps games involve many other options, such as “Come/Don’t Come Bets” and “Horn Bets.”

Other dice games are used for gambling, each with their own multiple versions and strategies. For example, in the dice game Ship, Captain, and Crew, a players gets three rolls of five dice to gain a ship (“6”), a captain (“5”), and a crew (“4”) in that order (or simultaneously). When those special numbers are rolled, that die is removed from play, with a successful player’s score being the sum of a roll of the two remaining dice.

In Buck Dice, a player throws one die to determine the “point number.” Another player then rolls three dice, continuing the rolls as long as one of the dice equals the point number. When this doesn’t occur, the player’s score for that round is the number of rolls. A“big buck” occurs when all three dice equal the point number, and the player withdraws from the game. A “little buck” occurs if all three die do not equal the point number, which adds 5 points to the player’s score. Any player with exactly 15 points withdraws from the game; any score forced higher than 15 nullifies a roll, and the player must reroll. The loser is the last person without reaching 15.

In Aces, a player starts with at least five dice, which he or she loses according to the numbers thrown. All rolled “1”s are placed in the table’s center and eliminated. All rolled “2”s are passed to the player on the left, while all “5”s are passed to the player on the right. Turns continue with rolls of the remaining dice until players either do not throw a “1,” “2,” or “5,” or have lost all of their dice. Play continues around the table until the last die rolled is a “1,” and the player who threw it is the winner.

Farkle begins with a player rolling six dice. Each “1” adds 100 points, each “5” adds 50 points, and if three dice show the same number, the player adds 100 times that shown number. A player can stop after any roll and keep the current total. Alternately, a player can roll again to possibly increase his or her score. But, if the next dice do not produce a positive score, the player lose all accumulated points for that round. The winner is the first to reach 10,000 points. Some variations of Farkle give 1000 points for shown runs of “1–5” or “2–6.”

In line with their history, multiple versions of dice games exist and will continue to be used by gamblers. Thus, the players who understand the probabilities involved will always have the advantage.

Bibliography

Barboianu, Catalin. Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery, and Sport Bets. Craiova, Romania: INFAROM Publishing, 2008.

Bell, R. C. Board and Table Games From Many Civilizations. New York: Dover Publications, 1979.

Devlin, Keith. The Unfinished Game. New York: Basic Books, 2008.

Mohr, Merilyn. The Game Treasury. Shelburne, VT: Chapters Publishing, 1993.