Distinct hands is the number of different ways to draw the hand, not counting different suits.

The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. In straight poker and five-card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Frequencies 5-card poker hands An Euler diagram depicting poker hands and their odds from a typical American 9/6 Jacks or Better machine To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. These interactions led to the conception of basic probability theory. Communicating through letters, the two continued to exchange their ideas and thoughts. Pascal's work on this problem began an important correspondence between him and fellow mathematician Pierre de Fermat (1601-1665).

Determined to know why his strategy was unsuccessful, he consulted with Pascal. De Méré tried a new mathematical approach to a gambling game but did not get the desired results. His friend, Chevalier de Méré, was an avid gambler with the goal to become wealthy from it.

Blaise Pascal (1623-1662) also contributed to probability theory. However, his work did not receive any immediate recognition since it was not published until after his death. His work from 1550, titled Liber de Ludo Aleae, discussed the concepts of probability and how they were directly related to gambling. Motivated by Paccioli's work, Girolamo Cardano (1501-1576) made further developments in probability theory. In 1494, Fra Luca Paccioli released his work Summa de arithmetica, geometria, proportioni e proportionalita which was the first written text on probability. The development of probability theory in the late 1400s was attributed to gambling when playing a game with high stakes, players wanted to know what the chance of winning would be.

Probability and gambling have been ideas since long before the invention of poker. 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.

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