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Three Card Poker Appendix 1
IntroductionI've been asked several times about how to calculate the probabilities in Three Card Poker. The math is much the same as for five card poker. However by popular demand I will show how I arrived at the probabilities in three card poker.
First, there are combin(52,3)=22100 ways to draw 3 cards out of 52, without regard to order. This is (52*51*50)/(3*2*1). For more information about the combin function visit my five card poker or lottery sections. The reason I use the combin notation is a habit developed from using Excel.
Straight Flush. There are 4 possible suits for the straight flush. The span of the straight flush may be from A23 to QKA, or 12 total spans. So the total number of straight flushes is 4*12=48.
Three of a kind. There are 13 possible ranks for the three of a kind. There are combin(4,3)=4 ways to choose 3 suits out of 4 within a rank. So there are 13*4=52 possible three of a kinds.
Straight. From the straight flush section we know there are 12 possible spans for a straight. A straight has 3 cards, each may be one of four suits. However if all three suits are the same then the player has a straight flush. So the number of suit combinations is 43-4 = 64-4 = 60. So there are 12*60=720 possible straights.
Flush. There are 4 possible suits for the flush. For each suit there are combin(13,3)=286 ways to draw 3 ranks out of 13. However we know from the straight flush section that there are 12 combinations which result in three connected ranks, giving the player a straight flush. So the combinations giving a straight, but not a straight flush, is combin(13,3)-12 = 286-12 = 274. So the number of flush combinations is 4*274=1096.
Pair. There are 13 possible ranks for the pair and 12 left for the singleton. So there are 13*12=156 ways to pick the ranks. Within the pair there are combin(4,2)=6 ways to pick 2 suits out of 4. For the rank of the singleton there are 4 possible suits. So the total suit combinations is 6*4=24. The total number of pair combinations is 156*24=3744.
Nothing. From the flush section we know there are 274 ways to pick 3 different ranks out of 13, without forming a straight. From the straight section we know there are 60 ways to pick 3 suits without forming a flush. So the number of ways to get nothing (less than a pair) is 274*60=16440.
To get the probability of any of these hands just divide the combinations by 22100.
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