Poker probability
In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.
People thought about probability and gambling long before the invention of poker. Gambling led to the development of probability theory in the late 1400s. When playing a game with high stakes, players wanted to know what the chance of winning would be. In 1494, Fra Luca Paccioli released his work Summa de arithmetica, geometria, proportioni e proportionalita which was the first written text on probability. Motivated by Paccioli’s work, Girolamo Cardano (1501-1576) made further developments in probability theory. His work from 1550, titled Liber de Ludo Aleae, discussed the concepts of probability and how they directly related to gambling. However, his work did not receive any recognition because it was not published until after his death. Blaise Pascal (1623-1662) also contributed to probability theory. His friend, Chevalier de Méré, was an avid gambler with the goal to become wealthy from it. De Méré tried a new mathematical approach to a gambling game but did not get the desired results. Determined to know why his strategy was unsuccessful, he consulted with Pascal. Pascal’s work on this problem began an important correspondence between him and fellow mathematician Pierre de Fermat (1601-1665). Communicating through letters, the two continued to exchange their ideas and thoughts. These interactions led to the conception of basic probability theory. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling.
The following chart enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. In this chart:
The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields as above.
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