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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

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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?

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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.

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Probabilities of Poker Hands

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2017 – Dan Ma

Texas hold ’em poker winning hands

The sequence of winning hands in poker is the single most important thing to know and memorise. Otherwise how will you know if you are winning the hand or on the losing end? This is important to understand if you want to make money in the casino or the reach the final table in a tournament. Sure, skill comes into it after that, and a Texas Hold’em winning poker strategy, but luck plays enough of a part that you can bypass those aspects if you recognise when you have a chance on the table. A good strategy for beginners Texas Holdem poker and part of learning how to get better at poker is to sit down and remember the types of hands below, the winning poker hands. For winning hands in Omaha poker check here.

Don’t forget that the best poker hand is made of a total five cards from any or all of the five cards on the table and the two in your hand. So the cards laid out below represent that, not just the cards on the table. So if you want to know how to be a better poker player this is the start.

Winning hand sequence; starting from the best,

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1: Royal Flush.

Ace, King, Queen, Jack, Ten all of one suit, ie diamonds or hearts or clubs or spades. A rare hand, the best hand in p0ker, statistically you are only likely to see one every 650,000 hands. If you do flop, turn or river a Royal Flush, you need a good strategy to get the most chips off other players. This is where you need the best poker tournament strategy you can get. Your play depends on how the other players on the table are betting. If they are loose and call everything, you can raise or even go all in and try and get them to call you. But if they are tight you need to play it wisely and let them make the running, re-raise them if you get the chance, draw them into your winning web.

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2; The straight flush.

Five cards all of the same suit, in sequence.

3; Four of a kind.

Known as quads, four cards of the same value. If you have one or two of them you will win. If there are four of the same cards on the board, whoever has the highest card to go with them will win, ie ace, and if two people have the highest card its a split pot.

4; Full house.

This hand consists of two cards of one value and three or another. If the three cards are kings and the two cards are sevens, its called Kings full. Whichever value cards is the most, they are the full hand. Any three / two card combination will do it.

5; The flush.

A flush is five or more cards of one suit. If two people have a flush the one with the hightest card in the flush wins the hand. And if you have ever wondered if a flush beats three of a kind, now you know, it does.

6; Straight.

For a straight you need five cards in sequence. They can be of any suit. 6,7,8,9,10 is an example. Any straight needs to have a five or a ten in it. Sometimes people get confused about what is better, a straight or a flush. Even though a straight seems harder to get than a flush, its the flush that wins over a straight, every tine.

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7; Three of a kind.

Three cards all the same rank.

8; Two pairs.

Any two pairs of two cards.

9; Pair.

Two cards of the same value, such as 77 or KK.

10; High card.

If no one makes a hand out of all the cards that come down and the cards in their hand, then the highest card will win. The best highest card is an ace, but it could be a four depending how the hand plays out.