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math 1040 five card stud probability

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  Andrew Christiansen Brandy Maestas Kelly Taylor Yuvia Hernandez Game:  5 card Stud Two Pair To find the probability of getting a two pair in a hand of cards you calculate ( 13 C 2 )( 4 C 2 )( 4 C 2 )( 11 C 1 )( 4 C 1 ) = 123,552 possible hands. To calculate the total possible number of hands you calculate ( 52 C 5 ) = 2,598,960. To find the probability of getting a two pair we divide the possible number of two pair hands by the total possible hands. So 123,552/2,598,960 = 0.047539. This means there is a 4.8% of drawing a hand with two pairs in it. Full House   →  (three cards showing the same number plus a pair) If ordered 5-card hand with the three-of-a-kind first, we have - 13 cards per suit, so 13 C 1  choices for the number showing on the first three cards. - Choice of suits, we have three out of four possible suits or 4 C 3  = 4 possibilities. Remaining pair, we have 12 C 1  choices for the number showing on the two cards. - For the choice of suits, we have two out of four possible suits or 4 C 2  = 6 possibilities. ➢   Full Houses = ( 13 C 1 )( 4 C 3 )( 12 C 1 )( 4 C 2 ) = 13·4·12·6 = 3744 ➢    Number of possible 5-card hands is 52 C 5  = 2,598,960 Dividing by the number of possible hands gives the probability: P(full house) = 3744 / 2,598,960 = 0.0014405762 = 0.144%  Four of a Kind The total number of possible hands you can draw can be found by calculating 52 nCr 5 = 2,598,960 hands. To get the total number of four of a kinds you could draw you calculate (13 ranks nCr 1 rank)*(4 suits nCr 4 suits)*(12 cards nCr 1 rank)*(4 suits nCr 1 suit) = 624 hands. ➢   52 C 5  = 2,598,960 hands ➢   ( 13 C 1 )( 1 2C 1 )( 4 C 1 )=624 hands The possibility of drawing four of a kind is found with 624/2,598,960 = 0.00024 or 0.024% Straight Flush  There are 10 straight flushes per suit and 4 suits, giving 40 possible straight flushes. {A-2  –  3  –  4  –  5 … 10 -J-Q-K-A} To get this probability, we count the number of possible straight flushes, and then divide by the number of all possible 5-card hands. ➢   52-5=47 ➢   The total number of five card hands is 52 C 5  = 52! / (47! 5!) = 2,598,960 ➢   The probability of getting dealt a straight flush in five cards is 40 / 2,598,960 = 1 / 64,974 = 1.5391E-05 or 0.0015391% Each player is dealt 5 cards highest hand wins. The rank of hands from lowest to highest is: 1.   high card   2.   pair   3.   4-straight   4.   4-flush   5.   two pair   6.   three of a kind   7.   straight   8.   flush   9.   full house   10.   four of a kind   11.   straight flush  
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