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

This means: in this specific example situation, the ICM advises a fold although the player has on average a much better hand than his opponent. Why is a fold better in this situation when the player is the clear favourite in the hand? Simply put: the short stack, player 4, is to blame. For player 3, it is much more profitable to wait for him to. In this video we teach you the hand rankings in Texas Hold'em - The most popular format of Poker. Get more Free Poker resources at http://www.pokervip.comLik.

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?

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

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2017 – Dan Ma

Below is the complete guide for determining how to rank various poker hands. This article covers all poker hands, from hands in standard games of poker, to lowball, to playing with a variety of wild cards. Scroll to the end to find an in-depth ranking of suits for several countries, including many European countries and North American continental standards.

Standard Poker Rankings

A standard deck of cards has 52 in a pack. Individually cards rank, high to low:

Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2

In standard poker (in North America) there is no suit ranking. A poker hand has 5 cards total. Higher ranked hands beat lower ones, and within the same kind of hand higher value cards beat lower value cards.

#1 Straight Flush

In games without wild cards, this is the highest ranking hand. It consists of five cards in sequence of the same suit. When comparing flushes, the hand with the highest value high card wins. Example: 5-6-7-8-9, all spades, is a straight flush. A-K-Q-J-10 is the highest ranking straight flush and is called a Royal Flush. Flushes are not permitted to turn the corner, for example, 3-2-A-K-Q is not a straight flush.

#2 Four of a Kind (Quads)

A four of a kind is four cards of equal rank, for example, four jacks. The kicker, the fifth card, may be any other card. When comparing two four of a kinds, the highest value set wins. For example, 5-5-5-5-J is beat by 10-10-10-10-2. If two players happen to have a four of a kind of equal value, the player with the highest ranking kicker wins.

#3 Full House (Boat)

A full house consists of 3 cards of one rank and 2 cards of another. The three cards value determines rank within Full Houses, the player with the highest rank 3 cards wins. If the three cards are equal rank the pairs decide. Example: Q-Q-Q-3-3 beats 10-10-10-A-A BUT 10-10-10-A-A would beat 10-10-10-J-J.

#4 Flush

Any five cards of the same suit. The highest card in a flush determines its rank between other flushes. If those are equal, continue comparing the next highest cards until a winner can be determined.

#5 Straight

Five cards in sequence from different suits. The hand with the highest ranking top card wins within straights. Ace can either be a high card or low card, but not both. The wheel, or the lowest straight, is 5-4-3-2-A, where the top card is five.

#6 Three of a Kind (Triplets/Trips)

A three of a kind is three card of equal rank and two other cards (not of equal rank). The three of a kind with the highest rank wins, in the event they are equal, the high card of the two remaining cards determines the winner.

#7 Two Pairs

A pair is two cards that are equal in rank. A hand with two pairs consists of two separate pairs of different ranks. For example, K-K-3-3-6, where 6 is the odd card. The hand with the highest pair wins if there are multiple two pairs regardless of the other cards in hand. To demonstrate, K-K-5-5-2 beats Q-Q-10-10-9 because K > Q, despite 10 > 5.

#8 Pair

A hand with a single pair has two cards of equal rank and three other cards of any rank (as long as none are the same.) When comparing pairs, the one with highest value cards wins. If they are equal, compare the highest value oddball cards, if those are equal continue comparing until a win can be determined. An example hand would be: 10-10-6-3-2

#9 High Card (Nothing/No Pair)

If your hand does not conform to any of the criterion mentioned above, does not form any sort of sequence, and are at least two different suits, this hand is called high card. The highest value card, when comparing these hands, determines the winning hand.

Low Poker Hand Ranking

In Lowball or high-low games, or other poker games which lowest ranking hand wins, they are ranked accordingly.

A low hand with no combination is named by it’s highest ranking card. For example, a hand with 10-6-5-3-2 is described as “10-down” or “10-low.”

Ace to Five

The most common system for ranking low hands. Aces are always low card and straights and flushes do not count. Under Ace-to-5, 5-4-3-2-A is the best hand. As with standard poker, hands compared by the high card. So, 6-4-3-2-A beats 6-5-3-2-A AND beats 7-4-3-2-A. This is because 4 < 5 and 6 < 7.

The best hand with a pair is A-A-4-3-2, this is often referred to as California Lowball. In high-low games of poker, there is often a conditioned employed called “eight or better” which qualifies players to win part of the pot. Their hand must have an 8 or lower to be considered. The worst hand under this condition would be 8-7-6-5-4.

Duece to Seven

The hands under this system rank almost the same as in standard poker. It includes straights and flushes, lowest hand wins. However, this system always considers aces as high cards (A-2-3-4-5 is not a straight.) Under this system, the best hand is 7-5-4-3-2 (in mixed suits), a reference to its namesake. As always, highest card is compared first. In duece-to-7, the best hand with a pair is 2-2-5-4-3, although is beat by A-K-Q-J-9, the worst hand with high cards. This is sometimes referred to as “Kansas City Lowball.”

Ace to Six

This is the system often used in home poker games, straights and flushes count, and aces are low cards. Under Ace-to-6, 5-4-3-2-A is a bad hand because it is a straight. The best low hand is 6-4-3-2-A. Since aces are low, A-K-Q-J-10 is not a straight and is considered king-down (or king-low). Ace is low card so K-Q-J-10-A is lower than K-Q-J-10-2. A pair of aces also beats a pair of twos.

In games with more than five cards, players can choose to not use their highest value cards in order to assemble the lowest hand possible.

Hand Rankings with Wild Cards

Wild cards may be used to substitute any card a player may need to make a particular hand. Jokers are often used as wild cards and are added to the deck (making the game played with 54 as opposed to 52 cards). If players choose to stick with a standard deck, 1+ cards may be determined at the start as wild cards. For example, all the twos in the deck (deuces wild) or the “one-eyed jacks” (the jacks of hearts and spades).

Wild cards can be used to:

• substitute any card not in a player’s hand OR
• make a special “five of a kind”

Five of a Kind

Five of a Kind is the highest hand of all and beats a Royal Flush. When comparing five of a kinds, the highest value five cards win. Aces are the highest card of all.

The Bug

Some poker games, most notably five card draw, are played with the bug. The bug is an added joker which functions as a limited wild card. It may only be used as an ace or a card needed to complete a straight or a flush. Under this system, the highest hand is a five of a kind of aces, but no other five of a kind is legal. In a hand, with any other four of a kind the joker counts as an ace kicker.

Wild Cards – Low Poker

During a low poker game, the wild card is a “fitter,” a card used to complete a hand which is of lowest value in the low hand ranking system used. In standard poker, 6-5-3-2-joker would be considered 6-6-5-3-2. In ace-to-five, the wild card would be an ace, and deuce-to-seven the wild card would be a 7.

Lowest Card Wild

Home poker games may play with player’s lowest, or lowest concealed card, as a wild card. This applies to the card of lowest value during the showdown. Aces are considered high and two low under this variant.

Double Ace Flush

This variant allows the wild card to be ANY card, including one already held by a player. This allows for the opportunity to have a double ace flush.

Natural Hand v. Wild Hand

There is a house rule which says a “natural hand” beats a hand that is equal to it with wild cards. Hands with more wild cards may be considered “more wild” and therefore beat by a less wild hand with only one wild card. This rule must be agreed upon before the deal begins.

Incomplete Hands

If you are comparing hands in a variant of poker which there are less than five cards, there are no straights, flushes, or full houses. There is only four of a kind, three of a kind, pairs (2 pairs and single pairs), and high card. If the hand has an even number of cards there may not be a kicker.

Examples of scoring incomplete hands:

10-10-K beats 10-10-6-2 because K > 6. However, 10-10-6 is beat by 10-10-6-2 because of the fourth card. Also, a 10 alone will beat 9-6. But, 9-6 beats 9-5-3, and that beats 9-5, which beats 9.

Ranking Suits

Poker Hands Ranking In Spanish Crossword

In standard poker, suits are NOT ranked. If there are equal hands the pot is split. However, depending on the variant of poker, there are situations when cards must be ranked by suits. For example:

• Drawing cards to pick player’s seats
• Determining the first better in stud poker
• In the event an uneven pot is to be split, determining who gets the odd chip.

Typically in North America (or for English speakers), suits are ranked in reverse alphabetical order.

• Spades (highest suit), Hearts, Diamonds, Clubs (lowest suit)

Poker Hands Ranking In Spanish Translation

Suits are ranked differently in other countries/ parts of the world:

• Spades (high suit), Diamonds, Clubs, Hearts (low suit)
• Hearts (high suit), Spades, Diamonds, Clubs (low suit) – Greece and Turkey
• Hearts (high suit), Diamonds, Spades, Clubs (low suit) – Austria and Sweden
• Hearts (high suit), Diamonds, Clubs, Spades (low suit) – Italy
• Diamonds (high suit), Spades, Hearts, Clubs (low suit) – Brazil
• Clubs (high suit), Spades, Hearts, Diamonds (low suit) – Germany

REFERENCES:

http://www.cardplayer.com/rules-of-poker/hand-rankings

https://www.pagat.com/poker/rules/ranking.html

Poker Hands Ranking In Spanish Language

https://www.partypoker.com/how-to-play/hand-rankings.html