The poker odds are in your favor. Now if your opponent bet $100 that would be a much dicier proposition. Suddenly you are risking $100 to win $200 and the odds are 200-100, or simplified, 2-1. Since you’re only 6-1 to make your hand you should probably fold. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands. Frequency of 7-card poker hands. In some popular variations of poker such as Texas Hold 'Em, a player uses the best five-card poker hand out of.

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.

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.

Best Poker Starting Hand Odds

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

Types of Starting Hands

by Rich McComas (updated Feb 6, 2004)

Below, I have categorized thirteen types of starting hands, in order of their value. The statistics are borrowed from www.pokalyzer.com which has ran 700 billion simulated hands to produce their results.

1. High Pairs (80%ers)

A high pair is a pair of Aces, Kings, or Queens. Some people count Jacks as high pairs, but I do not as they are not statistically matched to the value of other high pairs. My advice is Raise Pre-Flop and drive the garbage out. If you are in a late position, then you do not need to raise if someone else has driven out the garbage for you. However, if you are in the blind and someone else has raised, you might want to re-raise in order to increase the size of the pot from one more round of betting, and to drive out someone like a small blind who may have limped in with a mediocre hand. Whatever you do, DO NOT limp in with a high pair. If you fail to raise above the big blind and get beat by someone with a suited non-connector, it will be your own fault. If no one else has raised, it is your job to do so in every case.

These are legitimately the best hands in poker, so rather than jam the pot and re-raise pre-flop you should probably just call the raise, or slow-bet. If someone else has raised ahead of you, the garbage will be cleared and you can wait till the flop to reveal a betting strategy. If you flop a King and Ace, and you are holding two King, you will be in the driver's seat and no one will know what hit them.

ODDS: You are going to land any given pair every 220 hands you play, so a pocket pair of aces is extremely rare. Assuming you play 100 hands a night, it will only happen every other night. The chance of landing one of the high pairs, however, is once in every 73 hands, so it should happen to you once every evening. Don't lose out on that opportunity.

HAND	Rank	Win%
Pair of Aces	1	84.9%
Pair of Kings	2	82.1%
Pair of Queens	3	79.6%

2. Medium Pairs (70%ers)

A medium pair includes Jacks, Tens, Nines, Eights and Sevens. These are medium pairs because odds are that an overcard will flop more than half of the time. Even though the odds of winning with a pair of Jacks is 77.2%, the chances of having the high pair after the flop are only 43%. You are hoping for trips so, but if you don't land them, only continue if you have the high pair possible, and then bet high to throw out the single face cards in the hole. Do not be suckered by landing another board pair, giving you two pair. Either someone else has trips or someone with a face card will get the high pair on the turn or river.

ODDS: The chance of landing any pocket pair of sevens or higher is the same as landing two cards ten or higher. These events happen every 5.4 hands you play (or 16% of the time).

HAND	Rank	Win%
Pair of Jacks	4	77.2%
Pair of Tens	5	74.7%
Pair of Nines	6	71.7%
Pair of Eights	7	68.7%
Pair of Sevens	9	65.3%

3. Ace-Face Suited (65%ers)

An ace with another face card of the same suit is Ace-High suited, which is a winning hand most of the time. Two suited hole cards only draw to a flush 3% of the time, but if they are both high, you have a great shot at winning high pair as well. Because you hold the ace, any draw to a straight is a close-end straight, so your odds are lower than an open-end straight.

ODDS: You will receive two suited cards every 3.3 hands, and it is likely that at least two players at the table have suited cards, so don't get over-excited about just any suited combination.

HAND	Rank	Win%
AK Suited	8	66.2%
AQ Suited	10	65.3%
AJ Suited	11	64.4%

4. Ace-Face Offsuit (63%ers)

An Ace plus a face card of another suit are the most frequently played hands in Hold 'em. Most people stay in and raise with this hand. If you pair the face card, you will have the high kicker, and if you pair the ace, you will probably have a higher kicker than anyone else. If you draw to a straight, you will beat other straights and at least split the pot.

HAND	Rank	Win%
AK Offsuit	12	64.5%
AQ Offsuit	14	63.5%
AJ Offsuit	15	62.5%

5. Ace-Low Suited (60%ers)

An ace with anything 10 or lower in the same suit. This is not a hand to go heads up against someone with. This is best played in late position with a lot of callers already in the pot, giving you higher pot odds for a flush draw. This is also a good hand for stealing the blinds if you are on the button, because at least you have one ace in the hole. Mostly, though, I like to play this hand only in No Limit Hold'em because I may need to stay in to the river to get my cards, and I want to be able to go all-in and make a big score if I have the high flush. Earning a minor pot on such low odds just doesn't make as much sense.

POT ODDS: While the chances of winning against random hands are respectable, the chances of winning in heads-up play are not. Two suited pocket cards will flop a flush less than 1% of the time, and complete a flush by the river 6.52% of the time, and two separated cards (such as A5) will hit a draw will hit a draw only about 0.06% of the time. The probability of making a flush, therefore, is 15 to 1 against, meaning that you should really only enter the pot if there are 15 other players in, which never happens, so playing these cards with any regularity is a losing proposition.

If you are on the button, and the flop is cheap, and you do go with Ace-Low suited, then you should fold unless you immediately receive either two cards that match your hole cards or a four flush. The probability of flopping a four-flush is 8.1/1 or 10.9%, and the possibility of completing the flush by the river is 1.9/1 or 35%. When betting on the turn hoping for flush, you really want at least three people in the pot, or enough money for three in the pot.

HAND	Rank	Win%
AT Suited	13	63.5%
A8 Suited	21	60.5%
A6 Suited	30	58.2%
A4 Suited	35	57.1%
A2 Suited	46	55.5%

6. Face-Face Offsuit (58%)

Two face cards are best used when drawing to a straight, so you want to get into the flop as cheaply as possible. If your high card is a King, you will flop an Ace 23% of the time, and if your highest is a queen, a higher card will flop 41% of the time.

Poker Starting Hands Odds Calculator

HAND	Rank	Win%
KQ Offsuit	23	60.4%
KJ Offsuit	26	59.4%
QJ Offsuit	39	56.9%

7. Low Pairs (55%ers)

Low pairs are sixes or lower. All low pairs are questionable investments at best. If you all you have are have sixes, the chances are that someone will flop a higher pair 95% of the time. If you have a low pair and you flop a open-end straight draw, stay in, because you own TWO of the hole cards needed for the straight, lowering the odds that you will have to split the pot.

ODDS: You are going to land a pocket pair about once every 16 hands so don't get too excited about the lower pairs. Half the time, your pocket pairs will be high or medium pairs. Stick to those.

POT ODDS: Some people play low pairs in late position hoping to see trips, which is also the hope for higher pairs. If you take a pocket pair to the river, you have a 4.2/1 (19%) chance of making a set or better, so NEVER put money in the pot with this hand unless there are at least five other players. However, I don't bet on low pairs EVER, because it is more likely that someone else will take a straight or a flush or higher trips or a full house to the river and beat my low trips. With low pairs, I would never want to stay in unless I saw trips on the flop, and the chances of that are 7.5/1 (or 11.8%) so I would need 9 other people in the pot to make money, which is a very rare occurrence.

HAND	Rank	Win%
Pair of Sixes	17	62.7%
Pair of Fives	27	59.6%
Pair of Fours	48	56.3%
Pair of Threes	66	52.8%
Pair of Deuces	87	49.4%

8. King Flush Draw (55%ers)

A king plus another card in the same suit is a dangerous hand, unless the Ace is on the board, which will happen one-third of the time that you draw to a flush. These cards should be considered the same value as low pairs, except with a low pair, at least you know if you have trips on the flop. With a King Flush draw, you could have the ace on the flop, and still not get the flush on fifth street, so this is a potentially costly hand. See the section on 'Sucker Hands' for more info.

HAND	Rank	Win%
KQ Suited	16	62.4%
KT Suited	22	60.6%
K8 Suited	37	56.8%
K6 Suited	50	54.8%
K4 Suited	60	52.9%

9. Ace-Low Offsuit (55%ers)

An Ace plus a non-suited card lower than a face card is one of the biggest losers in Hold'em. Most players stay in with any ace, especially if you are playing with less than 10 at a table. However, I routinely fold this hand because 75% of the time, with 20 cards dealt into the pocket, someone else is ALSO holding an Ace in the pocket, and more often than not, their kicker will be higher than yours.

HAND	Rank	Win%
AT Suited	19	61.6%
A8 Suited	32	58.4%
A6 Suited	42	55.9%
A4 Suited	49	54.7%
A2 Suited	59	53.0%

10. Suited Connectors (45%ers)

Two suited cards next to each other are only slightly better than non-suited connectors because they help with the straight flush. If you draw to a flush or a straight ONLY, however, you are in serious risk of losing to larger straights or flushes. Don't get too excited about these cards because they share the same suit. The flush draw potential only increases your chances of winning by 3%. I generally fold all suited connectors, occasionally keeping a QJ or JT if I am in late position and several players are in and none have raised. If, after the flop, your hole card is at the low end of the straight, this is a classic 'Sucker Hand.' I know of some good players who love suited connectors, however, because they know on the flop if they have a killer hand, and a further investment will not be required. My attitude is that if you love suited connectors pre-flop, you might as well get equally excited about any low straight draw.

Poker Odds Starting Hands

ODDS: You will receive a suited connector once every 46 hands (or 2.1% of the time), and this is so rare, that people tend to jump for joy whenever it happens, not thinking that the changes of getting a 2-4 are even worse.

HAND	Rank	Win%
JT Suited	45	56.2%
98 Suited	83	48.9%
76 Suited	115	42.8%
54 Suited	136	38.5%
32 Suited	163	33.1%

11. Low Straight Draw (42%ers)

Two cards of different suits next to each other in value are going for a straight draw, and they are 3% less likely to win than the comparable suited connectors. Like suited connectors, only stay in if the pot is big because there are lots of other players calling pre-flop. Two low cards not in order is a Trash hand, and is not a Low Straight draw even if they are separated by only one card. Also, 3-2 in the pocket is the worst possible hand in poker even it if is a straight draw. An excellent analysis of unsuited connectors can be found at: http://www.pokerstove.com/unsuited.txt.

HAND	Rank	Win%
JT Offsuit	57	53.8%
98 Offsuit	99	46.1%
76 Offsuit	133	39.7%
54 Offsuit	153	35.0%
32 Offsuit	169	29.2%

12. Two Suited Cards (40%ers)

Two other suited cards, with at least one NOT being a face card, results in a four-flush flop only 10% of the time, and only a third of those finish as a flush. This is a loser hand.

HAND	Rank	Win%
T7 Suited	84	48.7%
T2 Suited	118	42.5%
83 Suited	139	38.3%
73 Suited	143	37.3%
62 Suited	156	32.8%

13. Trash Hands

Everything not mentioned above is a trash hand (as are some of the hands above, in my opinion), and there is no coincidence that these are in unlucky category 13. The only reason anyone stays in with these hands is that the flop was so cheap that the couldn't say no, such as being in the big blind with no raises. If you get a trash hand, and you are in the small blind, only consider calling if everyone else has and if you are sure that the person in the big blind won't raise you.