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Suit Combination
In the card game contract bridge, a suit combination is a specific subset of the cards of one suit held respectively in declarer's and dummy's hands at the onset of play. While the ranks of the remaining cards held by the defenders can be deduced precisely, their location is unknown. Optimum suit combination play allows for all possible lies of the cards held by the defenders. The term is also used for the sequence of plays from the declarer and dummy hands, conditional on intervening plays by the opponents; in other words, declarer's plan or strategy of play given his holdings and his goal for the number of tricks to be taken. In addition to understanding the possible initial combinations and probabilities for the location of the opponents' cards in a suit, declarer can further inform himself from the bidding, the opening lead and from the prior play of cards in establishing the probable location of remaining cards. Examples The diagram at left shows a heart suit combination ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Card Game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games (such as poker). A small number of card games played with traditional decks have formally standardized rules with international tournaments being held, but most are folk games whose rules vary by region, culture, and person. Traditional card games are played with a ''deck'' or ''pack'' of playing cards which are identical in size and shape. Each card has two sides, the ''face'' and the ''back''. Normally the backs of the cards are indistinguishable. The faces of the cards may all be unique, or there can be duplicates. The composition of a deck is known to each player. In some cases several decks are shuffled together to form a single ''pack'' or ''shoe''. Modern card games usually have bespoke decks, often with a vast amount of cards, and can include number or action cards. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacant Places
In the card game bridge, the law or principle of vacant places is a simple method for estimating the probable location of any particular card in the four hands. It can be used both to aid in a decision at the table and to derive the entire suit division probability table. At the beginning of a deal, each of four hands comprises thirteen cards and one may say there are thirteen vacant places in each hand. The probability that a particular card lies in a particular hand is one-quarter, or 13/52, the proportion of vacant places in that hand. From the perspective of a player who sees one hand, the probable lie of a missing card in a particular one of the other hands is one-third. The principle of vacant places is a rule for updating those uniform probabilities as one learns about the deal during the auction and the play. Essentially, as the lies of some cards become known – especially as the entire distributions of some suits become known – the odds on location of any othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Safety Play
Safety play in contract bridge is a generic name for plays in which declarer maximizes the chances for fulfilling the contract (or achieving a certain score) by ignoring a chance for a higher score. Declarer uses safety plays to cope with potentially unfavorable layouts of the opponent's cards. In so doing, declarer attempts to ensure the contract even in worst-case scenarios, by giving up the possibility of overtricks. Safety plays adapt declarer's strategy to the scoring system. In IMP-scoring tournaments and rubber bridge, the primary scoring reward comes from fulfilling the contract and overtricks are of little marginal value. Therefore, safety plays are an important part of declarer technique at quantitative scoring. In matchpoint games, which use comparative scoring, overtricks are very important. Therefore, although safety plays have a certain role at matchpoints, they are normally avoided if the odds for making the contract are good and overtricks are likely. Definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Restricted Choice (bridge)
In contract bridge, the principle of restricted choice states that "The play of a card which may have been selected as a choice of equal plays increases the chance that the player started with a holding in which his choice was restricted." Crucially, it helps play "in situations which used to be thought of as guesswork." For example, South leads a low spade, West plays a low one, North plays the queen, East wins with the king. The ace and king are equivalent cards; East's play of the king decreases the probability East holds the ace – and increases the probability West holds the ace. The principle helps other players infer the locations of unobserved equivalent cards such as that spade ace after observing the king. The increase or decrease in probability is an example of Bayesian updating as evidence accumulates and particular applications of restricted choice are similar to the Monty Hall problem. In many of those situations the rule derived from the principle is to ''play ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finesse
In contract bridge and similar games, a finesse is a type of card play technique which will enable a player to win an additional trick or tricks should there be a favorable position of one or more cards in the hands of the opponents. The player attempts to win either the current trick or a later trick with a card of the suit he leads notwithstanding that the opponents hold a higher card in the suit; the attempt is based on the assumption that the higher card is held by a particular opponent. The specifics of the technique vary depending upon the suit combination being played and the number of tricks the player is attempting to win in that suit. Terminology To ''finesse a card'' is to play that card. Thus, in the example, the Queen is finessed. The outstanding King is the card finessed ''against'', or the card the player hopes to capture by the finessing maneuver. Thus, you finesse against a missing honor, but you finesse the card you yourself play, the card finessed being s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bridge Probabilities
In the game of bridge mathematical probabilities play a significant role. Different declarer play strategies lead to success depending on the distribution of opponent's cards. To decide which strategy has highest likelihood of success, the declarer needs to have at least an elementary knowledge of probabilities. The tables below specify the various prior probabilities, i.e. the probabilities in the absence of any further information. During bidding and play, more information about the hands becomes available, allowing players to improve their probability estimates. Probability of suit distributions (for missing trumps, etc.) in two hidden hands This table"Mathematical Tables" (Table 4). represents the different ways that two to eight particular cards may be distributed, or may ''lie'' or ''split'', between two unknown 13-card hands (before the bidding and play, or ''a priori''). The table also shows the number of combinations of particular cards that match any numerical split an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. "Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Strategy
In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines. The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep their's unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Strategy
In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines. The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duplicate Bridge
Duplicate bridge is a variation of contract bridge where the same set of bridge deals (i.e. the distribution of the 52 cards among the four hands) are played by different competitors, and scoring is based on relative performance. In this way, every hand, whether strong or weak, is played in competition with others playing identical cards, and the element of skill is heightened while that of chance is reduced. This stands in contrast to Bridge played without duplication, where each hand is freshly dealt and where scores may be more affected by chance in the short run. Four-way card holders known as Bridge boards are used to enable each player's hand to be preserved from table to table, and final scores are calculated by comparing each pair's result with others who played the same hand. In duplicate bridge, players normally play all the hands with the same partner, and compete either as a partnership (in a 'Pairs tournament') or on a team with one or more other partnerships ('Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
