Module 2 · Lesson 7 of 39 · Free
The ten hand ranks and how often they happen
Of the 2,598,960 different five-card hands, only 4 are royal flushes, A K Q J T of one suit. Hold'em gives you seven cards to make your five from, and even then a royal flush comes about once in 30,940 deals. Every rank follows the same rule: the rarer a hand, the higher it ranks.
The hand
You hold A♥ K♥, and the board is Q♥ J♥ T♥ 4♣ 2♠. Your best five, A♥ K♥ Q♥ J♥ T♥, is a royal flush: the top of the ladder, and no hand can beat it.
Predict first
With five cards, which comes more often: a flush or a straight?
- A flush: five cards of one suit are easy to get
- A straight: there are almost twice as many
- Both come equally often
Show the answer
Answer: A straight: there are almost twice as many
Of the 2,598,960 five-card hands, 10,200 are straights and only 5,108 are flushes: 10,200 ÷ 5,108 ≈ 2. The flush is the rarer hand, so a flush beats a straight.
Working it out
Read the ladder from the bottom. Of the 2,598,960 five-card hands, 1,302,540 are high card only and 1,098,240 are one pair, so more than nine hands in ten hold a pair at most. Higher up, the hands thin out fast: there are 5,108 flushes, one hand in 2,598,960 ÷ 5,108 ≈ 509, and 4 royal flushes, one in 2,598,960 ÷ 4 = 649,740. In Hold'em your hand is the best five of seven cards: your two hole cards and the five on the board. Of the 133,784,560 different sets of seven, 4,324 hold a royal flush, one in 133,784,560 ÷ 4,324 = 30,940. Seven cards lift most hands: only 23,294,460 of the sets give nothing better than high card, so 133,784,560 − 23,294,460 = 110,490,100 of them, 82.6%, give one pair or better.
A few rules complete the ladder. The ace plays high or low in a straight, so A K Q J T is the highest straight and 5 4 3 2 A the lowest, but the ace cannot sit in the middle: Q K A 2 3 is no straight, only ace high. Suits never rank: two flushes with the same five ranks tie, whatever their suits. The order is set by five-card hands, because a hand is always five cards: with seven cards, ending up with only high card is even rarer than two pair, yet it still ranks lowest.
Formula
One hand in how many = all hands ÷ hands of the rank: a royal flush is 2,598,960 ÷ 4 = 649,740 with five cards, 133,784,560 ÷ 4,324 = 30,940 with seven
Check your understanding
You hold a flush. Which of these hands beats yours?
- A full house
- A straight
- Three of a kind
Show the answer
Answer: A full house
Of these three, only the full house is rarer than a flush: 3,744 five-card hands against 5,108. A straight, 10,200, and three of a kind, 54,912, are more common, so your flush beats them.
With five cards, 624 of the 2,598,960 hands are four of a kind. On average, one hand in how many is four of a kind?
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Answer: 4,165
2,598,960 ÷ 624 = 4,165: one five-card hand in 4,165.
Which of these five cards make a straight?
- Q♠ K♦ A♣ 2♥ 3♠
- 5♠ 4♦ 3♣ 2♥ A♠
- K♠ Q♦ J♣ 9♥ 8♠
Show the answer
Answer: 5♠ 4♦ 3♣ 2♥ A♠
The ace plays low as well as high, so 5 4 3 2 A is a straight, the lowest one. The ace cannot sit in the middle, so Q K A 2 3 is only ace high, and K Q J 9 8 has no ten to join it up.
With seven cards, 17.4% of deals give only a high card and 43.8% give exactly one pair. What share give two pair or better, in %?
Show the answer
Answer: 38.8%
Every other deal gives two pair or better: 100 − 17.4 − 43.8 = 38.8%.