Creation Date: 7 Jan 2005 | |
Last Modified: 11 Jun 2005 |
Various odds in Poker.
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*(4^{5} - 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 | 4^{5} [13!/(5! 8!)] - 4*[13!/(5! 8!) - 4^{5}*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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