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.

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).

(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).

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:

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

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!)

A couple of interesting pieces of trivia:

There are 4,234 ways to make a Royal Flush from any seven cards (1,081 ways in each suite; There are five essential cards that must be part of a Royal Flush: Ten, Jack, Queen, King, Ace. Of the remaining 47 cards, any two others can be selected, so the frequency of all Royal Flushes is 4C1 × 47C2 = 4,324). We are totally agnostic about what these other two cards are; they can be anything at all.

There are, however, nine flavors of Straight Flush (excluding the Royal Flush), straights with the top card: King, Queen, Jack, Ten, Nine, Eight, Seven, Six, Five (Ace can be used as low in poker) . There are 37,260 possible ways to make these Straight Flushes (9,315 per suit). There are nine possible high cards (9C1) multiplied by four possible suits (4C1), and of the remaining cards there are two we need to pick, but we must make sure we're not picking the card directly above the top card, so there are (46C2) to select from.

For each of these Straight Flushes, there is a poison card (but a good kind of poison). Neither of the two unused cards can be this poison card. For example, imagine we have a Nine-high Straight Flush in Hearts. We don’t care what the other two cards are, unless, one of these cards is the Ten of Hearts, if it is, it would not be a Nine-high Straight, but a Ten-high Straight. Similarly, if we had a King-high Straight Flush, and one of the other two cards happens to be the Ace, then we would have a Royal Flush, not a Straight Flush (Remember, we made the assumption that the ‘cards speak’, and the player automatically gets the best hand possible from the seven cards; why settle for a lower Straight Flush, when you can have a higher one? Why settle for a Straight Flush, when you can have a Royal Flush!)

It’s not a paradox after all, just math!