Poker Hand Rankings And Odds
This post works with 5-card Poker hands drawn from a standard deck of 52 cards. The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities
Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here). There are 2,598,960 many possible 5-card Poker hands. Thus the probability of obtaining any one specific hand is 1 in 2,598,960 (roughly 1 in 2.6 million). The probability of obtaining a given type of hands (e.g. three of a kind) is the number of possible hands for that type over 2,598,960. Thus this is primarily a counting exercise.
- As a poker player, knowing poker hand odds and rankings is crucial to knowing where you stand when calculating your odds of winning. This guide is for players from beginner to intermediate level – meaning those with a basic knowledge of poker but who don’t know how best to calculate poker odds to gauge the chances of success – and will give you everything you need to beat others when.
- Poker Hand Rankings - Texas Holdem Starting Hands Chart At the bottom of this page is a comprehensive listing of Texas Hold'em starting hands based on their EV (expected value). Expected value is the average number of big blinds this hand will make or lose.
Not only are the hand rankings modified but so are the mathematics and odds/probabilities of the majority of hands. Before we talk about the odds and probabilities of some of the hands, let’s have a look at the hand rankings offered in Six Plus Hold’em (ranked from the highest hand to the lowest): Six Plus Hold’em Hand Rankings Comparison.
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Preliminary Calculation
Usually the order in which the cards are dealt is not important (except in the case of stud poker). Thus the following three examples point to the same poker hand. The only difference is the order in which the cards are dealt.
These are the same hand. Order is not important.
The number of possible 5-card poker hands would then be the same as the number of 5-element subsets of 52 objects. The following is the total number of 5-card poker hands drawn from a standard deck of 52 cards.
The notation is called the binomial coefficient and is pronounced “n choose r”, which is identical to the number of -element subsets of a set with objects. Other notations for are , and . Many calculators have a function for . Of course the calculation can also be done by definition by first calculating factorials.
Thus the probability of obtaining a specific hand (say, 2, 6, 10, K, A, all diamond) would be 1 in 2,598,960. If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of all diamond cards? It is
This is definitely a very rare event (less than 0.05% chance of happening). The numerator 1,287 is the number of hands consisting of all diamond cards, which is obtained by the following calculation.
The reasoning for the above calculation is that to draw a 5-card hand consisting of all diamond, we are drawing 5 cards from the 13 diamond cards and drawing zero cards from the other 39 cards. Since (there is only one way to draw nothing), is the number of hands with all diamonds.
If 5 cards are randomly drawn, what is the probability of getting a 5-card hand consisting of cards in one suit? The probability of getting all 5 cards in another suit (say heart) would also be 1287/2598960. So we have the following derivation.
Thus getting a hand with all cards in one suit is 4 times more likely than getting one with all diamond, but is still a rare event (with about a 0.2% chance of happening). Some of the higher ranked poker hands are in one suit but with additional strict requirements. They will be further discussed below.
Another example. What is the probability of obtaining a hand that has 3 diamonds and 2 hearts? The answer is 22308/2598960 = 0.008583433. The number of “3 diamond, 2 heart” hands is calculated as follows:
One theme that emerges is that the multiplication principle is behind the numerator of a poker hand probability. For example, we can think of the process to get a 5-card hand with 3 diamonds and 2 hearts in three steps. The first is to draw 3 cards from the 13 diamond cards, the second is to draw 2 cards from the 13 heart cards, and the third is to draw zero from the remaining 26 cards. The third step can be omitted since the number of ways of choosing zero is 1. In any case, the number of possible ways to carry out that 2-step (or 3-step) process is to multiply all the possibilities together.
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The Poker Hands
Here’s a ranking chart of the Poker hands.
The chart lists the rankings with an example for each ranking. The examples are a good reminder of the definitions. The highest ranking of them all is the royal flush, which consists of 5 consecutive cards in one suit with the highest card being Ace. There is only one such hand in each suit. Thus the chance for getting a royal flush is 4 in 2,598,960.
Royal flush is a specific example of a straight flush, which consists of 5 consecutive cards in one suit. There are 10 such hands in one suit. So there are 40 hands for straight flush in total. A flush is a hand with 5 cards in the same suit but not in consecutive order (or not in sequence). Thus the requirement for flush is considerably more relaxed than a straight flush. A straight is like a straight flush in that the 5 cards are in sequence but the 5 cards in a straight are not of the same suit. For a more in depth discussion on Poker hands, see the Wikipedia entry on Poker hands.
The counting for some of these hands is done in the next section. The definition of the hands can be inferred from the above chart. For the sake of completeness, the following table lists out the definition.
Definitions of Poker Hands
| Poker Hand | Definition | |
|---|---|---|
| 1 | Royal Flush | A, K, Q, J, 10, all in the same suit |
| 2 | Straight Flush | Five consecutive cards, |
| all in the same suit | ||
| 3 | Four of a Kind | Four cards of the same rank, |
| one card of another rank | ||
| 4 | Full House | Three of a kind with a pair |
| 5 | Flush | Five cards of the same suit, |
| not in consecutive order | ||
| 6 | Straight | Five consecutive cards, |
| not of the same suit | ||
| 7 | Three of a Kind | Three cards of the same rank, |
| 2 cards of two other ranks | ||
| 8 | Two Pair | Two cards of the same rank, |
| two cards of another rank, | ||
| one card of a third rank | ||
| 9 | One Pair | Three cards of the same rank, |
| 3 cards of three other ranks | ||
| 10 | High Card | If no one has any of the above hands, |
| the player with the highest card wins |
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Counting Poker Hands
Straight Flush
Counting from A-K-Q-J-10, K-Q-J-10-9, Q-J-10-9-8, …, 6-5-4-3-2 to 5-4-3-2-A, there are 10 hands that are in sequence in a given suit. So there are 40 straight flush hands all together.
Four of a Kind
There is only one way to have a four of a kind for a given rank. The fifth card can be any one of the remaining 48 cards. Thus there are 48 possibilities of a four of a kind in one rank. Thus there are 13 x 48 = 624 many four of a kind in total.
Full House
Let’s fix two ranks, say 2 and 8. How many ways can we have three of 2 and two of 8? We are choosing 3 cards out of the four 2’s and choosing 2 cards out of the four 8’s. That would be = 4 x 6 = 24. But the two ranks can be other ranks too. How many ways can we pick two ranks out of 13? That would be 13 x 12 = 156. So the total number of possibilities for Full House is
Note that the multiplication principle is at work here. When we pick two ranks, the number of ways is 13 x 12 = 156. Why did we not use = 78?
Flush
There are = 1,287 possible hands with all cards in the same suit. Recall that there are only 10 straight flush on a given suit. Thus of all the 5-card hands with all cards in a given suit, there are 1,287-10 = 1,277 hands that are not straight flush. Thus the total number of flush hands is 4 x 1277 = 5,108.
Straight
There are 10 five-consecutive sequences in 13 cards (as shown in the explanation for straight flush in this section). In each such sequence, there are 4 choices for each card (one for each suit). Thus the number of 5-card hands with 5 cards in sequence is . Then we need to subtract the number of straight flushes (40) from this number. Thus the number of straight is 10240 – 10 = 10,200.
Three of a Kind
There are 13 ranks (from A, K, …, to 2). We choose one of them to have 3 cards in that rank and two other ranks to have one card in each of those ranks. The following derivation reflects all the choosing in this process.
Two Pair and One Pair
These two are left as exercises.
High Card
The count is the complement that makes up 2,598,960.
The following table gives the counts of all the poker hands. The probability is the fraction of the 2,598,960 hands that meet the requirement of the type of hands in question. Note that royal flush is not listed. This is because it is included in the count for straight flush. Royal flush is omitted so that he counts add up to 2,598,960.
Probabilities of Poker Hands
| Poker Hand | Count | Probability | |
|---|---|---|---|
| 2 | Straight Flush | 40 | 0.0000154 |
| 3 | Four of a Kind | 624 | 0.0002401 |
| 4 | Full House | 3,744 | 0.0014406 |
| 5 | Flush | 5,108 | 0.0019654 |
| 6 | Straight | 10,200 | 0.0039246 |
| 7 | Three of a Kind | 54,912 | 0.0211285 |
| 8 | Two Pair | 123,552 | 0.0475390 |
| 9 | One Pair | 1,098,240 | 0.4225690 |
| 10 | High Card | 1,302,540 | 0.5011774 |
| Total | 2,598,960 | 1.0000000 |
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2017 – Dan Ma
Most Commonly Asked Poker Questions
Not sure what beats a full house or what a straight can beat? Here are the answers to the most commonly-asked poker questions this side of the Strip.
Does a flush beat a full house?
No. A full house beats a flush in the standard poker hand rankings. The odds against making a full house in a game of Texas Hold’em are about 36-to-1, while the odds against making a flush are 32-to-1. The full house is a more rare hand and beats a flush.
Does a flush beat a straight?
Yes. Using the standard poker hand rankings, a flush beats a straight, regardless of the strength of the straight. The odds against making a straight in Texas Hold’em are about 21-to-1, making it a more common hand than a flush (32-to-1 odds against).
Does a straight beat a full house?
No. The odds against making a full house in Texas Hold’em are about 36-to-1, while the odds against making a straight are about 21-to-1. Both are strong five-card hands, but a full house occurs less often than a straight. A full house beats a straight in the poker hand rankings.
Does three of a kind beat two pair?
Yes. Both three of a kind and two pair can make a lot of money in poker, but three of a kind is the best hand when it goes head to head with two pair. The odds against making three of a kind in Texas Hold’em is about 20-to-1, while the odds against making two pair is about 3-to-1.
Does three of a kind beat a straight?
No. The odds of making both of these hands are very close in a game of Texas Hold’em. The odds against making a straight are 20.6-to-1, while the odds against making three of a kind are 19.7-to-1. The straight comes about slightly less often, making it the winner against three of a kind in the poker hand rankings.
Does a flush beat three of a kind?
Yes. The battle of strong hands between a flush and three of a kind sees the flush as the stronger hand. The odds against making a flush in Texas Hold’em are about 32-to-1, with odds against making three of a kind at around 20-to-1.
Does a straight beat two pair?
Yes. The poker hand rankings dictate that a straight is a stronger hand than two pair. The straight occurs with about 21-to-1 odds against in Texas Hold’em, while the odds against making two pair stand at about 3-to-1.
Does four of a kind beat a full house?
Yes. Both four of a kind and a full house are among the strongest poker hands, but four of a kind is a much rarer holding. Texas Hold’em odds against making four of a kind are 594-to-1, while you have about 36-to-1 odds against making a full house.
Does three of a kind beat a flush?
No. When the flush and three of a kind go head to head, the flush comes out as the best according to the poker hand rankings. The odds against making three of a kind sit around 20-to-1, with the odds against hitting a flush at 32-to-1.
Does a full house beat a straight in poker?
Yes. The full house comes in less often than a straight. In Texas Hold’em, the odds against drawing a full house are around 36-to-1, while the odds against making a straight are around 21-to-1.
Does a straight flush beat four of a kind?
Poker Hand Rankings And Odds Nfl
Yes. Four of a kind is an exceedingly rare hand in poker, but the straight flush is an even more elusive five-card hand. The odds against making a straight flush in Texas Hold’em is about 3,590-to-1, much rarer than four of a kind (594-to-1 odds against)