Why hand rankings follow rarity
In standard poker, the ranking of hands is largely a ranking by how rare they are. A hand that is dealt less often beats one that is dealt more often. That is not a coincidence: the order was set so that scarcer hands are worth more. This article looks at those frequencies, and at how common different two-card starting hands are in Texas hold’em.
Knowing the frequencies helps you judge what a hand is worth. It does not tell you what any particular hand will do, and it does not make a game profitable.
The odds for a five card hand
There are 2,598,960 different five-card hands from a 52 card deck. Counting how many of them fall into each category gives the standard table below (the counts are textbook results, and the methods for deriving them belong with five card hands counting).
| Hand | Number of hands | Approximate chance |
|---|---|---|
| Royal flush | 4 | 0.00015% |
| Straight flush (not royal) | 36 | 0.0014% |
| Four of a kind | 624 | 0.024% |
| Full house | 3,744 | 0.14% |
| Flush (not straight flush) | 5,108 | 0.20% |
| Straight (not straight flush) | 10,200 | 0.39% |
| Three of a kind | 54,912 | 2.1% |
| Two pair | 123,552 | 4.8% |
| One pair | 1,098,240 | 42.3% |
| High card only | 1,302,540 | 50.1% |
Check the total: 4 + 36 + 624 + 3,744 + 5,108 + 10,200 + 54,912 + 123,552 + 1,098,240 + 1,302,540 = 2,598,960. The categories are exclusive, so each hand is counted once, in its highest category.
Two points stand out. First, about half of all five-card hands contain nothing better than a high card, and over nine in ten (92.4%, adding 42.3% and 50.1%) are either a single pair or nothing. Second, the strongest hands are a tiny share. A full house or better means 4 + 36 + 624 + 3,744 = 4,408 hands out of 2,598,960, which is about 0.17%, or roughly 1 in 590.
Hold’em starting hands
In Texas hold’em each player starts with two cards, and there are 1,326 possible combinations of two cards. They divide into three groups:
- Pairs: 13 ranks × 6 combos = 78 hands, which is 78/1326 = 1/17, about 5.9%.
- Suited, different ranks: 78 rank pairs × 4 suits = 312 hands, about 23.5%.
- Offsuit, different ranks: 78 rank pairs × 12 = 936 hands, about 70.6%.
The three groups add to 78 + 312 + 936 = 1,326.
Specific hands are rarer. A particular pair, such as aces, has 6 combos, so 6/1326 = 1/221, about 0.45%. A particular suited hand such as ace-king of the same suit has 4 combos, 4/1326, about 0.30%. Ace-king in any suits has 16 combos, 16/1326, about 1.2%.
The practical lesson is that most starting hands are unremarkable, and the premium ones come up only occasionally. Over a long session you will see many weak hands and few strong ones, which is simply what the counts predict.
How frequencies inform decisions
Frequencies are a baseline. They tell you how likely a hand is before you know anything else. Once more cards appear, or once an opponent acts, the relevant question changes, and the useful numbers become outs, pot odds and the opponent’s likely hand ranges.
Two cautions are worth stating clearly.
The table describes dealt hands, not final hands. In hold’em you combine your two cards with up to five shared ones, so the chance of ending with a given category is different from the five-card table. Do not apply these percentages to the best five of seven cards without recalculating.
Rare does not mean unbeatable. A hand can be uncommon and still lose to a rarer one. Rarity is also relative to what others hold: your hand is only as strong as it is against the hands you might be facing.
Frequencies are not a prediction
A 42.3% chance of one pair means that, as the law of large numbers describes, across a very large number of hands about 42 in 100 are a single pair. In ten hands you may see none, or five. The probabilities do not “catch up”. After a long run without a strong starting hand, the next deal is no more likely to be strong, which is the gambler’s fallacy in action, and each shuffle is independent.
Poker is a mix of chance and decisions, and probability can improve the decisions, but it cannot remove the chance. Outcomes remain uncertain, money can be lost, and anyone who finds gambling hard to control should contact a local support service.