Counting Poker Hands
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1 Counting Poke Hands Geoge Ballinge In a standad deck of cads thee ae kinds of cads: ce (),,,,,,,,,, ack (), ueen () and ing (). Each of these kinds comes in fou suits: Spade (), Heat (), Diamond () and Club (). Thee ae cads altogethe: poke hand consists of an unodeed selection of five cads chosen fom a standad deck of cads. Thee ae C(, ) =,,0 distinct poke hands. Poke hands belong to one of ten categoies anging fom the highest anking, a oyal flush, to the lowest anking, a high cad. We descibe these categoies and use counting techniues to count the numbe of possible hands belonging to each categoy. Finally, we calculate the pobability of being dealt a -cad hand fom each categoy.. oyal flush consists of an ce, ing, ueen, ack and of the same suit. Thee ae fou oyal flush hands:
2 . staight flush is a consecutive seuence of five cads of the same suit, excluding a oyal flush. Fo each of the fou suits thee ae nine staight flushes. Fo example in heats the staight flushes ae: Thee ae = staight flushes in total.. fou of a kind consists of fou cads that ae all of the same kind togethe with a fifth cad of a diffeent kind. Examples of a fou of a kind poke hand ae: Thee ae C(, ) C(, ) ways of choosing one of the kinds and all fou suits followed then by C(, ) C(, ) ways of choosing one of the emaining kinds and one of the fou suits fo the fifth cad. By the poduct ule thee ae C(, ) C(, ) C(, ) C(, )= such hands.. full house consists of thee cads that ae all of the same kind (a thee of a kind ) togethe with two cads of anothe kind (a pai ). Examples of a full house ae: Thee ae C(, ) C(, ) ways of choosing one of the kinds and thee of the fou suits fo the thee of a kind followed then by C(, ) C(, ) ways of choosing one of the emaining kinds and two of the fou suits fo the pai. In total thee ae C(, ) C(, ) C(, ) C(, ) =, such hands.
3 . flush consists of five cads that ae all of the same suit, excluding a staight flush o a oyal flush. Examples of a flush ae: Thee ae C(, ) C(, ) ways of choosing five of the kinds and one of the fou suits. Howeve, since this also includes staight and oyal flushes, of which thee ae 0, then the total numbe of flushes is C(, ) C(, ) 0=,.. staight is a consecutive seuence of five cads that ae not all of the same suit. Examples of a staight ae: The smallest cad in a staight can be any of ten kinds: ce,,,,,,,, o. Including staight flushes and oyal flushes, each of the five cads can be any of the fou suits. Theefoe thee ae staights, staight flushes o oyal flushes. Finally, the total numbe of staights is 0=,00.. thee of a kind is a poke hand consisting of thee cads that ae all the same kind togethe with two cads of diffeent kinds. Examples of a thee of a kind ae: Thee ae C(, ) C(, ) ways of choosing one of the kinds and thee of the fou suits fo the thee of a kind followed then by C(, ) ways of choosing two of the emaining kinds and any of the fou suits fo the emaining two cads. ltogethe thee ae C(, ) C(, ) C(, ) =, such hands.
4 . two pai consists of five cads with two of one kind, two of a second kind and one of a thid kind. Examples of a two pai ae: Thee ae C(, ) C(, ) C(, ) ways of choosing two of the kinds and any two of the fou suits fo each of these two kinds to poduce two pais. Then thee ae C(, ) C(, ) ways of choosing one of the emaining kinds and one of the fou suits to make up the fifth cad. Theefoe thee ae C(, ) C(, ) C(, ) C(, ) C(, )=, such hands.. one pai consists of five cads whee two ae of the same kind and the othe thee ae of diffeent kinds. Examples of a one pai ae: Thee ae C(, ) C(, ) ways of choosing one of the kinds and any two of the fou suits fo the one pai. Then thee ae C(, ) ways of choosing thee of the emaining kinds and any of the fou suits fo each of these othe thee cads. So in total thee ae C(, ) C(, ) C(, ) =,0,0 such hands.. high cad is any poke hand that does not belong to one of the above nine categoies. In othe wods it consists of five cads, each of a diffeent kind, not all the same suit and not all in seuence. Examples of a high cad poke hand ae: Thee ae C(, ) ways of choosing five diffeent kinds of cads that ae not all in seuence and thee ae diffeent suits possible fo these five cads so they ae not all of the same suit. Thus in total thee ae (C(, ) ) ( )=,0,0 high cad hands.
5 The sum of the total numbes of hands fom each of these ten categoies is,,0, which is the total numbe of possible poke hands. Dividing each of these categoy totals by,,0 gives the pobability of being dealt a poke hand belonging to each categoy. The esults ae summaized in the following table. Categoy Sample Hand Numbe of Hands Pobability oyal flush staight flush fou of a kind full house, flush, 0.00 staight, thee of a kind, 0.0 two pai, 0.00 one pai,0,0 0.0 high cad,0,0 0.0 Total,,
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