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

1. Poker Straight Ace
2. Straight Poker King Ace Two Three Four Five A Straight
3. Poker Straight Ace Low
4. Straight Poker King Ace Two Three Four Seven

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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In ABC poker, a player willing to three- and four-bet has a premium hand, letting you know you're beat. This translates into wanting to conserve the losses and get out cheaply. Getting it all-in pre-flop in these situations will have you up against premium hands and coin flips most often, a weaker ace or king in occasional situations, and a 60. Poker is one of the many games involving the use of a 52-card deck of playing cards. The 52 cards are categorized by 13 ranks from Two through Ace (Aces can be counted as both higher than King and lower than Two when needed, but can only count as one at a time in a hand), and by four suits: diamonds, hearts, spades, and clubs.

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?

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

POKER GLOSSARY

All In

A name given to a play where a player stakes all their remaining chips on the hand that they are holding. Going all in was once considered to be the strongest indication that a player had a winning hand, but many players now go all in as a bluffing tactic.

Ante

An ante is a name that refers to a forced bid that all players have to place before they will be considered to be active within a hand. The ante is taken from the player’s account before cards are dealt and before the blinds are placed.

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Backdoor

The term backdoor is given when a player requires the turn and the river cards to transform their hand into a possible winning hand.

Backer

A backer is a person who sponsors a player to enter a poker tournament with a view to take a share of the player’s winnings should they place in the paying final positions.

Bad Beat

A bad beat refers to a hand that you lose after luck goes against you. A number of online poker rooms offer a new jackpot variation where they actually reward players with a large amount of money if they are subject to a bad beat.

Bankroll

The amount of funds that a poker player has at their disposal. The ultimate aim of poker playing is to increase your bankroll by as much as possible over your initial investment.

Bet

A bet is when a player wagers an amount of chips into the central pot to signal their hand strength.

Bicycle

A bicycle is a term that is used to describe a straight that is made up of A-2-3-4-5. This hand makes up the best possible hand in Low Ball games.

Blind

A blind is the name given to a forced bet that 2 players have to place before the dealer commences the game. The players who have to place the blinds are determined by the place of the dealer button.

Broadway

Name given to an ace high straight. The ace high straight would be made up of A-K-Q-J-10.

Bubble

The bubble is a name given to a player who is eliminated in the last non-paying place within a poker tournament. The loss of the bubble means that all players who are still remaining within the tournament will finish with a cash prize.

Bust

The term bust has a duel meaning. One meaning is used to name a hand that has failed to hit a card needed to empower the hand. The secondary meaning would be that a player is eliminated from a tournament or poker game.

Button

The button is an indicator for which player is considered to be the phantom dealer. This button is used to signal which players are to be included in the blinds. When a player is on the button they are considered to be in the best position because they get to see the majority of the table act before deciding how they want to proceed.

Buy In

A buy in is an amount of money that is charged to enter into a poker tournament or game in order for the player to be considered active in the game. The money collected from the buy in fees are used to create a final prize pool to reward players who reach the paying positions.

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Call

Name given to an amount of chips that are placed that are equal to either the blinds or a bet that has been made by another player. This action signals that you wish to remain active in the hand.

Chase

If a player was to chase a card they would be looking to stay within a hand even though the possibility of hitting the card that they require is poor.

Chase Card

A card that a player needs to chase in order to alter their hand. An example of this would be if a player was holding a set of 7s, the chase card would be the final 7 that is available in the deck.

Check

A signal that a player is happy to proceed with the hand without making any further bet. This play could be used in a number of ways, either to show strength through a check raise or weakness to continue the game without having to stake anymore into the central pot.

Check Raise

A check raise is a play that involves a player checking the action to another player in the hope that there will be a bet placed into the pot. Once the action comes back round to the player they will then place a raise into the pot. This is used to signal great strength.

Cold Call

A name given to a situation where a player decides that they want to call the wager in order to remain within the game but by doing so they are calling more than one bet. This could be that the big blind is raised and then the player calls the bet, so in theory the player is calling the big blind plus the further raise.

Come Hand

A hand that needs at least one additional card to become strong enough to win the hand.

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Dead Hand

A name given to a hand that fails to present an opportunity to win a hand after having an initial chance. The term tends to be used with hands that have fallen foul to the cards that are placed onto the community board. A great example of this would be a low pocket pair that has seen five over cards placed onto the community board.

Dealer

The name of the person who deals the cards for the game and does not factor into the game in any other way.

Draw

Name of a poker variant game where players discard some of the cards that are initially dealt to them at the start of the game. The players then draw cards from the deck.

Drawing Dead

A term used to describe when a player is in possession of a hand that can’t win the hand regardless of the cards that are still to be dealt.

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Family Pot

A name that is used when describing a pot where all or most of the players seated at the poker table are active.

Fixed Limit

A betting structure that is used across a number of poker variants that are indicated as limit poker. This means that players can only stake the amount that is set by the level of stakes that they have opted to play.

Flat Call

When a player just calls a bet when a raise would be seen as a more appropriate play.

Flop

The name given to the cards that the dealer reveals after the first round of betting concludes. The flop makes up the first three cards that are placed onto the community board.

Fold

Name given to the play that sees a player lay down their cards to signal that they no longer want to participate in the currently active game.

Forced Bet

This is an amount of chips that have to be placed into the central pot by a player in order to be part of the game. There are a number of reasons that a player might have to place a forced bet into the pot such as antes or when they are within the blind locations.

Freeroll

A style of poker tournament that offers tournament style game play to players without the need for the players to invest any funds into buy in. These tournaments tend to offer a low level prize pool.

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Gut Shot

A straight draw featuring four nearly consecutive cards with one hole such as a 4-5-7-8. Unlike open-ended straights, there is only one numerical card that can complete a gut shot.

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Heads Up

Term given to the section of a game where two players go head to head over a hand.

Hi Lo

Another variant of poker that alters the way that the game is played. The highest and lowest hand shares the central pot, meaning there are two winners of the pot in the game.

Hole Cards

The name given to the two cards that are dealt to a player by the dealer and that remain hidden and face down on the table.

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Jackpot

A prize fund that is offered by either the casino or online poker room that is rewarded once a player successfully completes a challenge that they set. An example of this would be the Bad Beat Jackpot offered by a number of online poker rooms and casinos where they reward players who suffer a bad beat above a set hand strength.

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Kicker

The name given to the card that a player has in their hand that accompanies a card that offers value into a hand. This is the card that will determine if the player wins a hand should their strong card be equaled by another player.

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Limp

A name given to a bet that is placed that only equals the value of the big blind. This is a tactic that is used to try to see another card at the cheapest possible price.

Loose

A term that is used when trying to describe a player’s approach to a game. The characteristics of a loose player would be that they play a lot of hands and bet aggressively while playing.

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Move in

Another name for an All In play that is made by a player, where the player stakes all of their remaining stack on the current hand.

Muck

Term used to name the process where a player discards their hand and fails to reveal their cards to anyone.

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No Limit

Name of a poker game structure where players don’t have any limits on the number of chips that they can wager into the pot. The no limit game of poker is the structure of game that offers some of the biggest central pots in poker history.

Nuts

Term that is used to describe the best possible hand that a player could be holding at that moment in the game. Players who are holding the nuts hand tend to slow play their opponents with a hope that they will come against a player who raises the stakes so that they can counter their play and increase their winnings.

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Offsuit

Name given to cards that are not of the same card suit. An example would be that an ace of clubs would be offsuit from king of diamonds.

Opener

The first player to act in a hand is referred to as the opener.

Over Call

Term used to describe when a player calls after another player has already called.

Over Card

Name given to a card within a player’s hand that is of higher value then the cards shown on the community board.

Over Pair

Name given to a player’s pocket pairs that offer a greater value than a pair that features on the community board.

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Pay Off

Name that refers to when a player makes the decision to make a call on the final round of betting even though they feel that they might be beaten.

Play the Board

Term that means that a player is using the 5 cards that are placed on the community board.

Position

Name that is used in reference to when a player acts. This is dictated by the rotation of the dealer button. The later the position the better that it is considered for the player to make an informed decision as to how to approach the hand.

Pot Limit

Betting structure where the players betting stakes are limited to a maximum of the value of the central pot.

Pot Odds

Name given to the percentages that players have to try to work out to ensure that they have a high chance of creating a winning hand.

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Qualifier

Name given to a set of requirements that need to be met to be able to qualify within a hand or to enter into a tournament.

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Rag

Name given to a low card that partners an ace. An example of this would be a card to the value of 6 or lower that is unsuited.

Rail Bird

Another name for a spectator who is watching from behind the rails.

Rainbow

A term used in the poker variant Texas Hold’em where the flop offers cards of different suits so that there is no possibility of players having a flush.

Raise

A bet that is used to signal hand strength and increase the central pot.

Rake

The amount of money that is taken by the online poker room from each hand. The amount that the casino or poker room take is dependent on the size of the central pot at the time of the hand coming to an end.

Read

The ability of being able to take information from a player and to create an expectation of what your opponent is holding within their hand. This can help a player to successfully win a hand either by having a better hand or by being able to play a successful bluff.

Re-buy

A term used when a player enters a tournament and gets eliminated. Certain tournaments offer a re-buy that means that players can buy into the tournament again at the original buy in cost.

Represent

Term used for when a player tries to be deceptive towards their opponents by acting in a way that would give their opponents the impression that they are in possession of a strong hand.

Ring Game

A game that operates using normal casino rules, where the chips that are taken to the table symbolize real money, as opposed to a tournament where each player is given a set number of chips for a set buy in.

River

Name given to the final card that is dealt by the dealer and placed onto the community board. This card has been known to make or break a huge amount of hands in Texas Hold’em where the term is used.

Runner

Term used to describe when a player manages to get two cards that they need to empower their hand on the final two cards to be dealt onto the community board in a game of Texas Hold’em.

Rush

Name that describes a run of good hands or good luck that a player might experience.

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Satellite

A style of tournament that offers a first-place prize of entry into a bigger tournament.

Scare Card

A card that is deal into play by the dealer that has the potential to be used to give the impression that you are holding a hand that contains that card. The name is given as this could lead to opponents folding if they feel their hand has now become less superior.

Scoop

Term used to describe when a player wins a full pot. This is common during Hi Low games where the pot is often split.

Set

Name for three of a kind.

Shootout

Tournament structure where players are placed onto tables and maintain to play on that table until there is only one player remaining and they possess all of the active chips on the table. The players are then reseated with other winning players to fight it out for the prize pool.

Side Pot

A pot that is created in case one of the active players cannot match the bets of other players. The remaining funds are placed into a side pot and are contested by only players who can have placed into the side pot.

Slow Play

A style of play that means a player plays their hand as if it was of a lesser value that it really is in the hope that an opponent takes the lead to raise the betting stakes.

Streets

Name given to the cards as they are dealt. The cards are named by how they were dealt from the deck so the river card would be called 5th street as it is the 5th card to be dealt onto the community board.

Suck Out

Name given to when a card that has entered into a pot significantly behind in terms of pot odds gets a lucky card that gives the player the stronger hand and they win the pot.

Suited Connectors

Name given to cards that are held within a player’s hole cards that are of the same suit and one value apart. An example of this would be a 7 and 8 of diamonds.

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Table Stakes

The amount that the table has set for the size of the blinds. These set the minimum bets that players can place.

Tell

Name given to signs that players give off to their opponents while they are playing that could offer some form of information into the hand that the player is holding.

Poker Straight Ace

Tight

This is a description of how a poker player might approach a game, a player who is tight will be less willing to make bluffs and make plays with hands that they feel are weak. These players tend to wait for power hands before they are willing to engage in a betting war with other players.

Tilt

A term used to describe when a player takes a loss of a hand badly and begins to lose their poker playing mentality causing them to lose focus and so causing them to make poor poker plays that will dent their chip stack.

Tournament

A game that features a greater number of players than a ring game and includes a bigger prize pool too. Poker rooms and other tournament organizers typically charge players entry fees to play in tournaments, which are used to fund the prize pool. These are the most publicized poker games in the world to date as they make great television viewing.

Turn

Name given to the 4th card that is dealt onto the community board. Also known as 4th Street.

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Value

There are two meanings to this term. The first meaning is the strength of the hand that a player is holding. The higher the value of their hand the higher it is seen in the poker hand rankings. The second meaning for the term is when a player plays a hand that has a slight chance of winning the hand but only slightly higher than the possibility of the hand falling to a higher strength hand.

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Straight Poker King Ace Two Three Four Five A Straight

Walk

Poker Straight Ace Low

Name given to when a poker player wins a pot without being contested by any player. An example is that all players fold while you are seated in the big blind.

Straight Poker King Ace Two Three Four Seven

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