Counting Poker Hands
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1 Counting Poker Hands George Ballinger In a standard deck of cards there are kinds of cards: ce (),,,,,,,,,, ack (), ueen () and ing (). Each of these kinds comes in four suits: Spade (), Heart (), Diamond () and Club (). There are cards altogether: poker hand consists of an unordered selection of five cards chosen from a standard deck of cards. There are C(, ) =,,0 distinct poker hands. Poker hands belong to one of ten categories ranging from the highest ranking, a royal flush, to the lowest ranking, a high card. We describe these categories and use counting techniques to count the number of possible hands belonging to each category. Finally, we calculate the probability of being dealt a -card hand from each category.. royal flush consists of an ce, ing, ueen, ack and of the same suit. There are four royal flush hands:
2 . straight flush is a consecutive sequence of five cards of the same suit, excluding a royal flush. For each of the four suits there are nine straight flushes. For example in hearts the straight flushes are: There are = straight flushes in total.. four of a kind consists of four cards that are all of the same kind together with a fifth card of a different kind. Examples of a four of a kind poker hand are: There are C(, ) C(, ) ways of choosing one of the kinds and all four suits followed then by C(, ) C(, ) ways of choosing one of the remaining kinds and one of the four suits for the fifth card. By the product rule there are C(, ) C(, ) C(, ) C(, )= such hands.. full house consists of three cards that are all of the same kind (a three of a kind ) together with two cards of another kind (a pair ). Examples of a full house are: There are C(, ) C(, ) ways of choosing one of the kinds and three of the four suits for the three of a kind followed then by C(, ) C(, ) ways of choosing one of the remaining kinds and two of the four suits for the pair. In total there are C(, ) C(, ) C(, ) C(, ) =, such hands.
3 . flush consists of five cards that are all of the same suit, excluding a straight flush or a royal flush. Examples of a flush are: There are C(, ) C(, ) ways of choosing five of the kinds and one of the four suits. However, since this also includes straight and royal flushes, of which there are 0, then the total number of flushes is C(, ) C(, ) 0=,.. straight is a consecutive sequence of five cards that are not all of the same suit. Examples of a straight are: The smallest card in a straight can be any of ten kinds: ce,,,,,,,, or. Including straight flushes and royal flushes, each of the five cards can be any of the four suits. Therefore there are straights, straight flushes or royal flushes. Finally, the total number of straights is 0=,00.. three of a kind is a poker hand consisting of three cards that are all the same kind together with two cards of different kinds. Examples of a three of a kind are: There are C(, ) C(, ) ways of choosing one of the kinds and three of the four suits for the three of a kind followed then by C(, ) ways of choosing two of the remaining kinds and any of the four suits for the remaining two cards. ltogether there are C(, ) C(, ) C(, ) =, such hands.
4 . two pair consists of five cards with two of one kind, two of a second kind and one of a third kind. Examples of a two pair are: There are C(, ) C(, ) C(, ) ways of choosing two of the kinds and any two of the four suits for each of these two kinds to produce two pairs. Then there are C(, ) C(, ) ways of choosing one of the remaining kinds and one of the four suits to make up the fifth card. Therefore there are C(, ) C(, ) C(, ) C(, ) C(, )=, such hands.. one pair consists of five cards where two are of the same kind and the other three are of different kinds. Examples of a one pair are: There are C(, ) C(, ) ways of choosing one of the kinds and any two of the four suits for the one pair. Then there are C(, ) ways of choosing three of the remaining kinds and any of the four suits for each of these other three cards. So in total there are C(, ) C(, ) C(, ) =,0,0 such hands.. high card is any poker hand that does not belong to one of the above nine categories. In other words it consists of five cards, each of a different kind, not all the same suit and not all in sequence. Examples of a high card poker hand are: There are C(, ) ways of choosing five different kinds of cards that are not all in sequence and there are different suits possible for these five cards so they are not all of the same suit. Thus in total there are (C(, ) ) ( )=,0,0 high card hands.
5 The sum of the total numbers of hands from each of these ten categories is,,0, which is the total number of possible poker hands. Dividing each of these category totals by,,0 gives the probability of being dealt a poker hand belonging to each category. The results are summarized in the following table. Category Sample Hand Number of Hands Probability royal flush straight flush four of a kind full house, flush, 0.00 straight, three of a kind, 0.0 two pair, 0.00 one pair,0,0 0.0 high card,0,0 0.0 Total,,
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