Dice are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance.
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric). If and , then (transitive).
A random variable is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data. A statistical model represents, often in considerably idealized form, the data-generating process. When referring specifically to probabilities, the corresponding term is probabilistic model. All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory".
In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.
Pig is a simple dice game first described in print by John Scarne in 1945. Players take turns to roll a single die as many times as they wish, adding all roll results to a running total, but losing their gained score for the turn if they roll a 1.
In mathematics, intransitivity is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.
Liar's dice is a class of dice games for two or more players requiring the ability to deceive and to detect an opponent's deception. In "single hand" liar's dice games, each player has a set of dice, all players roll once, and the bids relate to the dice each player can see plus all the concealed dice. In "common hand" games, there is one set of dice which is passed from player to player. The bids relate to the dice as they are in front of the bidder after selected dice have been re-rolled. Originating during the 15th century, the game subsequently spread to Latin American and European countries. In 1993, a variant, Call My Bluff, won the Spiel des Jahres.
Cee-lo is a gambling game played with three six-sided dice. There is not one standard set of rules, but there are some constants that hold true to all sets of rules. The name comes from the Chinese Sì-Wŭ-Liù (四五六), meaning "four-five-six". In America it is also called "See-Low," "Four-Five-Six," "The Three Dice Game," "Roll-off!," and by several alternative spellings, as well as simply "Dice." In China it is also called "Sān Liù Bàozi" (三六豹子), or "Three-Six Leopards". In Japan, it is known as "Chinchiro" (チンチロ) or "Chinchirorin" (チンチロリン).
In graph theory, a tournament is a directed graph with exactly one edge between each two vertices, in one of the two possible directions. Equivalently, a tournament is an orientation of an undirected complete graph; however, as directed graphs, tournaments are not complete: complete directed graphs have two edges, in both directions, between each two vertices. The name tournament comes from interpreting the graph as the outcome of a round-robin tournament, a game where each player is paired against every other exactly once. In a tournament, the vertices represent the players, and the edges between players point from the winner to the loser.
In game theory, strategic dominance occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.
Dice notation is a system to represent different combinations of dice in wargames and tabletop role-playing games using simple algebra-like notation such as d8+2.
Poker dice are dice which, instead of having number pips, have representations of playing cards upon them. Poker dice have six sides, one each of an Ace, King, Queen, Jack, 10, and 9, and are used to form a poker hand.
Bradley Efron is an American statistician. Efron has been president of the American Statistical Association (2004) and of the Institute of Mathematical Statistics (1987–1988). He is a past editor of the Journal of the American Statistical Association, and he is the founding editor of the Annals of Applied Statistics. Efron is also the recipient of many awards.
Penney's game, named after its inventor Walter Penney, is a binary (head/tail) sequence generating game between two players. Player A selects a sequence of heads and tails, and shows this sequence to player B. Player B then selects another sequence of heads and tails of the same length. Subsequently, a fair coin is tossed until either player A's or player B's sequence appears as a consecutive subsequence of the coin toss outcomes. The player whose sequence appears first wins.
An intransitive or non-transitive game is a term sometimes used for a (zero-sum) game in which pairwise competitions between the strategies contain a cycle. If strategy A beats strategy B, B beats C, and C beats A, then the binary relation "to beat" is intransitive, since transitivity would require that A beat C. The terms "transitive game" or "intransitive game" are not used in game theory, however.
Miwin's Dice are a set of nontransitive dice invented in 1975 by the physicist Michael Winkelmann. They consist of three different dice with faces bearing numbers from one to nine; opposite faces sum to nine, ten or eleven. The numbers on each die give the sum of 30 and have an arithmetic mean of five.
In economics, and in other social sciences, preference refers to an order by which an agent, while in search of an "optimal choice", ranks alternatives based on their respective utility. Preferences are evaluations that concern matters of value, in relation to practical reasoning. Individual preferences are determined by taste, need, ..., as opposed to price, availability or personal income. Classical economics assumes that people act in their best (rational) interest. In this context, rationality would dictate that, when given a choice, an individual will select an option that maximizes their self-interest. But preferences are not always transitive, both because real humans are far from always being rational and because in some situations preferences can form cycles, in which case there exists no well-defined optimal choice. An example of this is Efron dice.
Stochastic transitivity models are stochastic versions of the transitivity property of binary relations studied in mathematics. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments of paired comparisons, specifically in scenarios where transitivity is expected, however, empirical observations of the binary relation is probabilistic. For example, players' skills in a sport might be expected to be transitive, i.e. "if player A is better than B and B is better than C, then player A must be better than C"; however, in any given match, a weaker player might still end up winning with a positive probability. Tightly matched players might have a higher chance of observing this inversion while players with large differences in their skills might only see these inversions happen seldom. Stochastic transitivity models formalize such relations between the probabilities and the underlying transitive relation.
Given a pair of dice, and , let and be defined as the random variables which represent the sum of rolls of and correspondingly. Consider the arithmetic function for that indicates which dice has a higher probability of rolling a higher sum for rolls. For example, can be defined as if , if and otherwise.
