# Texas Hold'em Poker

I’m going to start out with a seemingly paradoxical statement: When playing Texas Hold’em Poker you are more likely to get a Royal Flush than a King-High Straight Flush!
How can that possibly be? There's only one of every card! The key to this mystery is that hands in Texas Hold’em are generated from selecting the best five out of seven cards. The frequency of occurrence of some hands is covertly modified by the two unused cards (but more of this later).
Having seven, rather than five, cards to select from is the reason for the difference.
You are more likely to get a Royal Flush than a King-High Straight Flush!

## Poker Hands

A quick recap. Poker hands are ranked according to likelihood of them occurring. There are 52C5 possible five card hands (2,598,960 distinct hands) that can be dealt from a deck, and here is a table showing a breakdown of the frequency of occurrence of each of these ranked hands. The less frequently occurring hands (such as Full Houses and Flushes) rank higher than more common hands (such as Three of a Kind and Two Pair).
HandFrequency
Royal Flush4
Straight Flush136
Four of a Kind624
Full House3,744
Flush25,108
Straight210,200
Three of a Kind54,912
Two Pair123,552
One Pair1,098,240
High Card1,302,540
TOTAL2,598,960
1Excludes the Royal Flushes
2Excludes Straight Flushes, and Royal Flushes
(If you want to understand how these figures were derived, see this article on Poker Odds, or, if you are one of the crazy people who plays with wild cards, this article that takes into account the modified probabilities with various combinations of wild cards).

Image: Guts Gaming

## Texas Holdem Poker

In Texas Hold’em, you have two private hold cards, and there are five community cards. You are allowed to make your hand from any permutation of these seven cards (using both, one, or neither of your hold cards).
This means that there are considerably more discrete sets of cards to consider. There are 52C7 = 133,784,560 distinct ways to pull seven cards from a deck.
It is always assumed that a player is attempting to make the best possible hand they can from the seven dealt cards. If not, please come to my house to play poker (and bring a fat wallet!)
Here is a table of the frequency of occurrence of each of these hands from the set of all possible seven cards in Texas Hold'em:
HandFrequency
Royal Flush4,324
Straight Flush137,260
Four of a Kind224,848
Full House3,473,184
Flush24,047,644
Straight26,180,020
Three of a Kind6,461,620
Two Pair31,433,400
One Pair58,627,800
High Card23,294,460
TOTAL133,784,560

## Probabilities

Let’s take a quick detour to look at probabilities/odds, not just frequencies.
You see more higher scoring rankings in Texas Hold’em than simple five card poker because the additional cards allow more chances to get these hands. Below are tables of the probabilities for each of the hands (it’s obtained by dividing the frequency of occurrence of each hand by the denominator of all possible hands), for five card poker, and Texas Hold'em. Also included is an Odds Against column, which is the ratio the number of ways to fail to the ratio of ways to win = (1/Pr)−1:1

## Five Card Poker

HandFrequencyProbabilityOdds Against
Royal Flush40.000154%649,739:1
Straight Flush360.001385%72,192.33:1
Four of a Kind6240.024010%4,164:1
Full House3,7440.144058%693.167:1
Flush5,1080.196540%507.802:1
Straight10,2000.392465%253.8:1
Three of a Kind54,9122.112845%46.33:1
Two Pair123,5524.753902%20.035:1
One Pair1,098,24042.256903%1.366:1
High Card1,302,54050.117739%0.995:1
TOTAL2,598,960

## Texas Hold'em Poker

HandFrequencyProbabilityOdds Against
Royal Flush4,3240.003232% 30,939:1
Straight Flush37,2600.027851% 3,589.568:1
Four of a Kind224,8480.168067% 594:1
Full House3,473,1842.596102% 37.519:1
Flush4,047,6443.025494% 32.052:1
Straight6,180,0204.619382% 20.648:1
Three of a Kind6,461,6204.829870% 19.704:1
Two Pair31,433,40023.495536% 3.256:1
One Pair58,627,80043.822546% 1.282:1
High Card23,294,46017.411920% 4.743:1
TOTAL133,784,560
Draw poker can’t be modelled in tables like these as it depends on choices made by the player on what cards to keep, and which to discard (Do you keep a natural Flush you’ve been dealt, or discard some of your cards hoping for a Straight Flush? What person would break up a Full House hunting for Four of a Kind?, or throw away one-half of Two Pair!)