Poker Hand Table

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In poker, players form sets of five playing cards, called hands, according to the rules of the game. Each hand has a rank, which is compared against the ranks of other hands participating in the showdown to decide who wins the pot. In high games, like Texas hold 'em and seven-card stud, the highest-ranking hands.

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.

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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.

Blackjack Hand Table

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.

Hands

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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 HandDefinition
1Royal FlushA, K, Q, J, 10, all in the same suit
2Straight FlushFive consecutive cards,
all in the same suit
3Four of a KindFour cards of the same rank,
one card of another rank
4Full HouseThree of a kind with a pair
5FlushFive cards of the same suit,
not in consecutive order
6StraightFive consecutive cards,
not of the same suit
7Three of a KindThree cards of the same rank,
2 cards of two other ranks
8Two PairTwo cards of the same rank,
two cards of another rank,
one card of a third rank
9One PairThree cards of the same rank,
3 cards of three other ranks
10High CardIf no one has any of the above hands,
the player with the highest card wins

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Table

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.

Poker Hand Table


Probabilities of Poker Hands

Poker HandCountProbability
2Straight Flush400.0000154
3Four of a Kind6240.0002401
4Full House3,7440.0014406
5Flush5,1080.0019654
6Straight10,2000.0039246
7Three of a Kind54,9120.0211285
8Two Pair123,5520.0475390
9One Pair1,098,2400.4225690
10High Card1,302,5400.5011774
Total2,598,9601.0000000

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2017 – Dan Ma

Poker Range Calculator

Poker Hand Table


The Poker Hand Range Calculator calculates Texas Hold'em hand ranges from percentage values and vice-versa. All this online and free. Setup a poker range by selecting the hands in the poker hand calculator and share the link which is automatically generated for you.

Questions? Refer to our Poker Hand Calculator FAQ page.

Poker Hand Range Calculator


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Poker Starting Hand Table


AAAKsAQsAJsATsA9sA8sA7sA6sA5sA4sA3sA2s
AKoKKKQsKJsKTsK9sK8sK7sK6sK5sK4sK3sK2s
AQoKQoQQQJsQTsQ9sQ8sQ7sQ6sQ5sQ4sQ3sQ2s
AJoKJoQJoJJJTsJ9sJ8sJ7sJ6sJ5sJ4sJ3sJ2s
AToKToQToJToTTT9sT8sT7sT6sT5sT4sT3sT2s
A9oK9oQ9oJ9oT9o9998s97s96s95s94s93s92s
A8oK8oQ8oJ8oT8o98o8887s86s85s84s83s82s
A7oK7oQ7oJ7oT7o97o87o7776s75s74s73s72s
A6oK6oQ6oJ6oT6o96o86o76o6665s64s63s62s
A5oK5oQ5oJ5oT5o95o85o75o65o5554s53s52s
A4oK4oQ4oJ4oT4o94o84o74o64o54o4443s42s
A3oK3oQ3oJ3oT3o93o83o73o63o53o43o3332s
A2oK2oQ2oJ2oT2o92o82o72o62o52o42o32o22

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Poker Hand Calculator For Poker Ranges


To fully utilize the poker hand calculator, follow the steps below:

  • 1. You obviously need a poker room to play. Make sure to check out one of the best poker rooms, William Hill.

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  • 4. Enjoy more profitable poker!