 # All Math Considered

## The probabilities of poker 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.

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.

The Poker Hands

Heres 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

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
Lets 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 2s and choosing 2 cards out of the four 8s. 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

2017 Dan Ma

## Poker Hands Odds

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 playing online poker.

### How Odds Work

Poker odds give you the probability of winning any given hand. Higher odds mean a lower chance of winning, meaning that when the odds are large against you it ll be a long time until you succeed. They are usually displayed as a number to number ratio and indicate the potential return on investment; for example, odds of nine to one (9:1) means that for every \$1 wagered you ll be paid \$9.

Before you can begin to understand poker odds, you ll need to learn how to calculate outs . Outs are the cards that can help you improve your hand and make it more valuable than what you believe your opponent has. There are 52 cards in a deck and two of those will be in your hand when starting playing, with a further four cards exposed from the flop and turn. That means that of the 37 cards that remain unseen, there are 9 potential winning cards or outs . That equates to odds of 4:1 for getting one of the cards, or outs, you need.

### Poker Odds Chart

If you need to know some of the odds and probabilities of common poker hands, take a look at our poker odds chart to quickly learn which hands to play. You can either print it out and keep it to hand during a game, or calculate poker odds at a glance online.

### Common Poker Hands Odds

To help you become a better player, we ve listed the odds of common poker hands and situations that you re likely to see at the table. To learn more about key tactics and terms when playing poker, visit our poker school.

#### Open-ended straight draws (4.8:1)

An open-ended straight draw (OESD) is a straight draw that can be completed at either end. For example, holding 6, 7, 8 and 9 means either a 10 or 5 will complete the straight. There are eight outs: the four fives and the four tens. However, you should be aware that these odds presume that there is no possible flush on the board and that you re drawing to the best hand, which may not be the case.

#### Four to a flush (4.1:1)

A four-flush (flush draw) is a hand that is one card short of being a full flush. If your hole cards (which you are dealt at the beginning of a hand) are suited and there are two more of your suit on the board, this is a good hand, as it s rare that another player will have two hole cards of your suit. However, you should be cautious if you don t have the ace as this will lower your odds of winning.

#### Inside straight (10.5:1)

Also known as a gutshot, an inside straight is four cards that form a straight with one of the middle three cards missing. For example, 8, 9, 10, J and Q is a straight removing the 8 or Q makes it an outside straight, and removing the 9, 10 or J makes it an inside straight. Again, these odds assume that you re aiming to draw the best hand possible, but unless you use both of your hole cards to make the straight, that won t be the case.

#### One pair to two pair or trips (8.2:1)

A two pair is the seventh-best possible poker hand and is formed with two cards of the same value. For example, if you have J-T and you suspect an opponent holds a pair of aces, you have five outs to beat him: three tens and two jacks. However, this is based on your opponent not having AJ or AT, which can be a dangerous assumption.

#### Overcards (6.7:1)

Overcard refers to hole cards which are of higher rank than any other cards on the board. For example, if you re holding Ace and Queen and the flop comes 10, 8 and 6, there are two overcards on the board. Depending on what you believe your opponent has, you have six outs and odds of 6.7:1 but this only holds true if your assumption is correct.

#### Drawing to a set (22:1)

A player is drawing if they have an incomplete hand and require further cards to complete it. It is often a really far-fetched draw and rarely warrants playing. Generally, if you can t make an accurate deduction of your opponent s hands when drawing to a set, you should always assume they have one that threatens your own and fold to avoid losing.

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