A quick note before the math
Casino games are games of chance with a built-in house edge, and no strategy, system or timing changes the odds in the player’s favor over time. Most players lose money over time, and nobody should gamble money they cannot afford to lose. This article is about counting cards, not about winning. If gambling is causing you difficulty, a local support service can help.
The ranking is a ladder of rarity
From the strongest to the weakest, the standard ranking of five-card poker hands is: royal flush (the highest straight flush), straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair, and high card.
This order is not arbitrary tradition. In standard poker, a hand ranks higher because fewer of the possible five-card hands make it. The rest of this article shows how to count that.
How many five-card hands exist
A 52-card deck can be dealt into five-card hands in a number of ways that do not depend on the order of the cards. The count is the number of ways to choose 5 cards from 52, written C(52,5):
C(52,5) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 2,598,960
That is the total every other count is compared against. Because each hand is equally likely from a well-shuffled deck, the chance of any rank is simply its count divided by 2,598,960. The general idea of choosing groups without regard to order is covered in more depth under card combinations.
Counting each hand type
Here is how each count is built. Ranks are the 13 values (2 through ace) and each rank has 4 suits.
- Straight flush, including royal: There are 10 possible straights by rank (ace-low through ace-high) and 4 suits, so 40 hands. Four of those are royal flushes (ten through ace), leaving 36 other straight flushes.
- Four of a kind: Choose the rank (13), take all four cards, then choose any of the 48 remaining cards as the fifth: 13 × 48 = 624.
- Full house: Choose the rank of the three (13) and which 3 of its 4 suits (4), then choose a different rank for the pair (12) and which 2 of its 4 suits (6): 13 × 4 × 12 × 6 = 3,744.
- Flush: Choose a suit (4) and 5 of its 13 cards, C(13,5) = 1,287, giving 5,148. Remove the 40 straight flushes already counted: 5,108.
- Straight: 10 possible rank sequences, and each of the five cards can be any of 4 suits: 10 × 4^5 = 10,240. Remove the 40 straight flushes: 10,200.
- Three of a kind: Choose the rank (13) and 3 of 4 suits (4), then two other different ranks from the remaining 12 (C(12,2) = 66) with any suit for each (4 × 4): 13 × 4 × 66 × 16 = 54,912.
- Two pair: Choose two ranks for the pairs (C(13,2) = 78), 2 of 4 suits for each (6 × 6 = 36), and a fifth card of any of the other 44 cards: 78 × 36 × 44 = 123,552.
- One pair: Choose the paired rank (13) and its suits (6), then three other different ranks from 12 (C(12,3) = 220) with any suit each (4^3 = 64): 13 × 6 × 220 × 64 = 1,098,240.
- High card: Everything else. Five different ranks that are not a straight (1,277 rank sets) with suits that are not all the same (1,020 suit patterns): 1,277 × 1,020 = 1,302,540.
The full table
| Hand | Number of hands | Roughly 1 in |
|---|---|---|
| Royal flush | 4 | 649,740 |
| Other straight flush | 36 | 72,193 |
| Four of a kind | 624 | 4,165 |
| Full house | 3,744 | 694 |
| Flush | 5,108 | 509 |
| Straight | 10,200 | 255 |
| Three of a kind | 54,912 | 47 |
| Two pair | 123,552 | 21 |
| One pair | 1,098,240 | 2.4 |
| High card | 1,302,540 | 2.0 |
As a check, the counts add up to 2,598,960. About 50 percent of random five-card hands are high card only, and about 42 percent are exactly one pair. Everything else together is under 8 percent.
Why the counts explain the order
Notice that the order of the table follows the counts from smallest to largest. A flush is rarer than a straight, so it ranks higher. A full house is rarer than a flush, so it ranks higher still. This is the entire logic of the standard ranking: rarity.
The exception that proves the rule is that the ranking depends on the deck and the hand size. With three-card hands, there are only 22,100 possible hands, and a straight (720 hands) is rarer than a flush (1,096 hands), so games built on three cards commonly rank a straight above a flush. Games using wild cards, short decks or different hand sizes change the counts and so can change what is rare.
Ties and kickers
Two hands of the same type are separated by the ranks of their cards, and in many cases by leftover cards called kickers. Suits normally do not break ties in standard poker. Two players can therefore split a pot when their hands are equal.
Why rankings matter for casino games
Hand rankings determine which hands a pay table rewards, and how rare each hand is determines how often it can be paid. A game that pays a lot for a rare hand must pay little for common ones to avoid losing money on average, which links directly to the idea of expected value and to the built-in house edge. In video poker, the frequency of each hand is the engine under the pay table. In table poker variants, the same counts decide where the operator’s edge comes from.
These counts also show something about expectations. A hand that arrives roughly once in 700 deals will frequently not arrive in a session at all, which is a feature of basic probability and small samples, not a sign that a game is broken or “due”. Knowing that the ranking is simply a frequency table makes it easier to see why no hand is ever owed to anyone, and why outcomes remain uncertain no matter how well the hands are understood.