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Three Card Poker Appendix 1
Introduction
I'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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