Creation Date: 7 Jan 2005
Last Modified: 11 Jun 2005

Poker Odds

Various odds in Poker.


The Basic Odds

These are the probabilities that a five-card poker hand will show up. In just about any poker game, you'll have more information available than is given here. For example, in Holdem, 5- or 7-Card Stud, there will be a number of face-up cards on the table. In the usual version of 5-card draw, the opening bettor has at least a pair of jacks. This means you'll always have more information available than is assumed here, and so can adjust the odds accordingly.

Hand How We Get It Formula Number Fraction Odds Against
Royal Flush AKQJT in Suit, Four Suits 4 4 0.0000015 649739:1
Straight Flush In order, in suit, but not a Royal Flush. Assume A2345 is allowed. 9*4 36 0.0000139 72192.33:1
Four of a Kind 13 hands, plus 48 kickers for each 4 of a kind 13*48 624 0.0002401 4164:1
Full House Given four suits, there are 4 ways to make a given 3-of-a-kind, and 6 ways to make a given pair. Since we can't have 5-of-a-kind, and we've already considered 4-of-a-kind, once we have a given 3-of-a-kind, there are 12(*6) possible pairs to complete the hand. (4*13)*(6*12) 3,744 0.0014406 693.17:1
Flush In a given suit, there are 13!/(5! 8!) ways of picking 5 cards. However, 10 of these ways give us a Straight Flush or Royal Flush. And, of course, there are 4 suits. 4*[13!/(5! 8!) - 10] 5,108 0.0019654 507.80:1
Straight In the sequence A23456789TJQKA there are 10 possible 5-card straights. Each card can be one of four suits, except that they can't be all of the same suit, 'cause that's a Straight or Royal Flush 10*(45 - 4) 10,200 0.0039246 253.80:1
Three of a Kind For a given rank, there are 4 possible 3-of-a-kind (♠, ♣, ♠♣, ♠♣) There are, of course, 13 ranks. The other two cards can be anything not of the same rank, in either order [48!/(2! 46!)], except for the 12*6 possible pairs. (4*13)*(48*47/2 - 6*12) 54,912 0.0211285 46.33:1
Two Pair In each of the 13 ranks there are 6 possible pairs (♠, ♠, ♠♣, , ♣, ♣). Once we pick a possible pair, there are 6*12 no-matching pairs. The order of the pairs doesn't matter, hence the factor of 2. Finally, the fifth card can be any of the 44 in the deck that don't match either pair. 44*[(6*13)*(6*12)/2] 123,552 0.0475390 20.04:1
Pair In a given rank there are 6 possible pairs, as above. The other 3 cards can be anything, giving 48!/[3! 45!], less the possible 2 pair and 3-of-a-kind hands (6*13)*(48*47*46/6-12*6*44-4*12) 1,098,240 0.4225690 1.37:1
Junk Ignoring suits, there are 13!/(5! 8!) ways of picking 5 cards which don't pair, triple, or quadruple up. Each card can be one of four suits. Except that we don't count Flushes, Straights, or Straight/Royal Flushes 45 [13!/(5! 8!)] - 4*[13!/(5! 8!) - 45*10 + 40 1,302,540 0.5011774 1.00:1
All Hands 52 card deck taken 5 at a time where the order doesn't matter. 52!/(5! 47!) 2,598,960 1.0000000 N/A

References

I've derived these odds myself, but I've checked them against various references:


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