5CARD POKER HANDS(most recent edit: January 2, 2005)A SINGLE PAIRThis the hand with the pattern AABCD, where A, B, C and D are from the distinct "kinds" of cards: aces, twos, threes, tens, jacks, queens, and kings (there are 13 kinds, and four of each kind, in the standard 52 card deck). The number of such hands is (13choose1)*(4choose2)*(12choose3)*[(4choose1)]^3. If all hands are equally likely, the probability of a single pair is obtained by dividing by (52choose5). This probability is 0.422569.TWO PAIRThis hand has the pattern AABBC where A, B, and C are from distinct kinds. The number of such hands is (13choose2)(4choose2)(4choose2)(11choose1)(4choose1). After dividing by (52choose5), the probability is 0.047539.A TRIPLEThis hand has the pattern AAABC where A, B, and C are from distinct kinds. The number of such hands is (13choose1)(4choose3)(12choose2)[4choose1]^2. The probability is 0.021128.
A FULL HOUSEThis hand has the pattern AAABB where A and B are from distinct kinds. The number of such hands is (13choose1)(4choose3)(12choose1)(4choose2). The probability is 0.001441.
FOUR OF A KINDThis hand has the pattern AAAAB where A and B are from distinct kinds. The number of such hands is (13choose1)(4choose4)(12choose1)(4choose1). The probability is 0.000240.
A STRAIGHTThis is five cards in a sequence (e.g., 4,5,6,7,8), with aces allowed to be either 1 or 13 (low or high) and with the cards allowed to be of the same suit (e.g., all hearts) or from some different suits. The number of such hands is 10*[4choose1]^5. The probability is 0.003940. IF YOU MEAN TO EXCLUDE STRAIGHT FLUSHES AND ROYAL FLUSHES (SEE BELOW), the number of such hands is 10*[4choose1]^5  36  4 = 10200, with probability 0.00392465
A FLUSHHere all 5 cards are from the same suit (they may also be a straight). The number of such hands is (4choose1)* (13choose5). The probability is approximately 0.00198079. IF YOU MEAN TO EXCLUDE STRAIGHT FLUSHES, SUBTRACT 4*10 (SEE THE NEXT TYPE OF HAND): the number of hands would then be (4choose1)*(13choose5)4*10, with probability approximately 0.0019654.
A STRAIGHT FLUSHAll 5 cards are from the same suit and they form a straight (they may also be a royal flush). The number of such hands is 4*10, and the probability is 0.0000153908. IF YOU MEAN TO EXCLUDE ROYAL FLUSHES, SUBTRACT 4 (SEE THE NEXT TYPE OF HAND): the number of hands would then be 4*104 = 36, with probability approximately 0.0000138517.
A ROYAL FLUSHThis consists of the ten, jack, queen, king, and ace of one suit. There are four such hands. The probability is 0.00000153908.
NONE OF THE ABOVEWe have to choose 5 distinct kinds (13choose5) but exclude any straights (subtract 10). We can have any pattern of suits except the 4 patterns where all 5 cards have the same suit: 4^54. The total number of such hands is [(13choose5)10]* (4^54). The probability is 0.501177.
