Math of Poker - Basics
The game of poker is a card game played among two or more players for several rounds. There are several varieties of the game, but they all tend to have these aspects in common: The game begins with each player putting down money allocated for betting. During each round of play, players are dealt cards from a standard 52-card deck, and the goal of each player is to have the best 5-card hand at the table. Players keep their cards hidden, and each player makes bets on the strength of his or her cards. When the round is over, the cards are revealed, and the player with the best hand wins the round and the money that was bet during that round. The game is over when a single player has won all the money at the table.
Although poker contains elements of randomness and gambling, there is a substantial amount of skill involved in understanding probability and game theory. Poker strategy typically involves an application of these concepts to improve one's chance of winning. Over the long run of rounds and games, higher skilled players tend to win more.
It should be noted that one does not need to be interested in poker strategy to be interested in the mathematics involved in poker. There are many interesting mathematical observations to be made about a deck of cards and probabilities of events in poker. This page will explore these mathematical observations.
Contents
Standard Poker Deck
Poker is played with a standard, 52-card deck.
A standard playing card deck, also called a poker deck, contains 52 distinct cards.
These cards are divided into four suits:
Hearts and Diamonds are the two red suits. These are sometimes abbreviated as H and D. Clubs and Spades are the two black suits. These are sometimes abbreviated as C and S.
There are 13 ranks in each suit: an Ace, nine cards numbered \(2\) through \(10\), and three face cards—the Jack, the Queen, and the King.
The face cards are abbreviated as J, Q, and K. The Ace is abbreviated as A.
Each distinct card has a rank and a suit. For example, a distinct card is the King of Diamonds, and it is identified as K\(\color{red}{♦}\).
Once one understands how a poker deck is structured, one can investigate probabilities of certain events.
What is the probability that a Heart card is drawn from a shuffled poker deck?
There are \(13\) hearts in the poker deck, and there are \(52\) cards in total.
Let \(H\) be the event that a heart card is drawn from the shuffled poker deck. By probability by outcomes,
\[P(H)=\frac{13}{52}=\frac{1}{4}.\]
The probability to draw a heart is \(\frac{1}{4}.\ _\square\)
One aspect of the strategy of poker is to think about what cards you would need in order to win the game. If you know the probability that you will get a card that you need, then you will have a good understanding of what your chances are of winning.
You are playing a game of poker, and you have just been dealt the following hand of cards:
\[3♠, 6♠, 7♠, \text{J}♠, 5{\color{red}{♥}}.\]
You put the \(5\color{red}{♥}\) aside and ask to be dealt a new card. What is the probability that the next card dealt to you is a spade?
There are 13 spades in a deck of 52 cards. With the five cards dealt to you, there are now 9 spades left in a deck of 47 cards.
Let \(D\) be the event that a spade is drawn. By probability by outcomes,
\[P(D)=\frac{9}{47}.\]
The probability that the next card dealt to you is a spade is \(\boxed{\frac{9}{47}}.\ _\square\)
\[4{\color{red}\heartsuit}, 4\clubsuit, 8\clubsuit, 8\spadesuit, \text{K}{\color{red}\diamondsuit}\]
You are playing a game of poker, and you are dealt the above hand of cards from a shuffled standard poker deck.
You put the \(\text{K}\color{red}{\diamondsuit }\) aside and ask to be dealt a new card from the same deck. What is the probability that the next card dealt to you is a 4 or an 8?
Round your answer to three decimal places.
Note: Cards that are dealt to you are no longer in the deck. The \(\text{K}\color{red}{\diamondsuit }\) is put aside; it is not put back into the deck.
Poker Hands
Regardless of which variety of poker is being played, the hands of poker typically remain the same.
A poker hand is a combination of 5 cards drawn from a poker deck. Each hand is valued by its classification.
Poker hands are combinations rather than permutations. This means that the order of the cards does not matter. For example, each of the hands below is considered to be the same hand:
- (9♣, 10♠, 3♣, 8\(\color{red}♦\), Q♠)
- (3♣, 8\(\color{red}♦\), 9♣, 10♠, Q♠)
- (3♣, Q♠, 9♣, 10♠, 8\(\color{red}♦\)).
Using the binomial coefficient, one can calculate the total number of possible hands.
How many possible poker hands are there?
There are 52 cards in a poker deck, and a hand is a combination of 5 of those cards. Therefore, the number of possible poker hands is
\[\binom{52}{5}=2,598,960.\ _\square\]
Poker hands are put into classifications so that players can know how much their hand is worth. The following is a list of poker hand classifications, listed from the least valuable to the most valuable:
High Card: This type of hand is any hand that cannot be classified as one of the types below.
Example: (3♣, 8\(\color{red}♦\), 9♣, 10♠, Q♠)
One Pair: This type of hand consists of 2 cards of the same rank and 3 other cards of distinct ranks.
Example: (J♣, J\(\color{red}♥\), 5\(\color{red}♦\), 10♣, Q\(\color{red}♥\))
Two Pair: This type hand consists of 2 cards of the same rank, another 2 cards of the same rank, and a \(5^\text{th}\) card of a different rank.
Example: (2\(\color{red}♦\), 2♣, 7\(\color{red}♥\), 7\(\color{red}♦\), A♠)
Three of a Kind: This type of hand consists of 3 cards of the same rank, and 2 other cards of distinct ranks.
Example: (Q♣, Q\(\color{red}♦\), Q♠, K♠, 4♣)
Straight: This type of hand consists of 5 consecutive cards by value. The face cards are valued above the numbered cards in the order J, Q, K. The Ace card can represent the lowest valued card or the highest valued card, but it cannot represent both.
Example: (A♣, 2\(\color{red}♥\), 3\(\color{red}♦\), 4♣, 5♠)
Example: (10\(\color{red}♥\), J\(\color{red}♥\), Q♣, K♠, A♣)
Non-Example: (J\(\color{red}♥\), Q♠, K♠, A\(\color{red}♥\), 2♠)
Flush: This type of hand consists of 5 cards of the same suit.
Example: (3\(\color{red}♥\), 5\(\color{red}♥\), 6\(\color{red}♥\), 10\(\color{red}♥\), K\(\color{red}♥\))
Full House: This type of hand consists of 3 cards of the same rank and another 2 cards of the same rank.
Example: (7\(\color{red}♥\), 7\(\color{red}♦\), 7♠, 9\(\color{red}♦\), 9♣)
Four of a Kind: This type of hand consists of 4 cards of the same rank and another card.
Example: (J\(\color{red}♥\), J\(\color{red}♦\), J♣, J♠, 3♣)
Straight Flush: This type of hand is a straight and a flush at the same time.
Example: (5\(\color{red}♦\), 6\(\color{red}♦\), 7\(\color{red}♦\), 8\(\color{red}♦\), 9\(\color{red}♦\))
Royal Flush: A royal flush is the highest possible straight flush. It consists of cards of the ranks 10, J, Q, K, and A that are all of the same suit.
Example: (10♣, J♣, Q♣, K♣, A♣)
These classifications are mutually exclusive and exhaustive. If a hand meets the criteria for two classifications, then it is always classified as the higher of those classifications. For example, the hand (7\(\color{red}♥\), 7\(\color{red}♦\), 7♠, 9\(\color{red}♦\), 9♣) would always be classified as a full house; it would never be classified as three of a kind or one pair.
Probabilities of Poker Hands
Each of the 2,598,960 possible hands of poker is equally likely when dealt 5 cards from a standard poker deck. Because of this, one can use probability by outcomes to compute the probabilities of each classification of poker hand.
The binomial coefficient can be used to calculate certain combinations of cards. Then, the counting principles of rule of sum and rule of product can be used to compute the frequency of each poker hand classification. Then, the probability of each poker hand classification is simply its frequency divided by 2,598,960.
The probabilities calculated below are based on drawing 5 cards from a shuffled poker deck. The likelihood of each type of hand determines its value. The less likely the hand, the more it is worth. For example, a flush is always better than a straight because a flush is less likely than a straight when drawing 5 cards from a shuffled poker deck. Although different variants of poker involve different rules on drawing cards, these rankings are always used to determine the best hand. The hand classifications below are ordered from least value (most likely) to most value (least likely).
It is recommended that you try to compute these probabilities on your own before looking at the computations shown here. These classifications are ordered by their relative frequencies, but it is not recommended that you start with the High Card Hand computation, as it is more complicated than other computations. There is more than one way to arrive at the correct answer, so do not despair if your methodology is not the exact same.
Probability of High Card Hand
\[P(\text{High Card Hand})=\frac{1277}{2548}\approx 0.501177\]\[\text{High Card Hand Frequency}=\left[\vphantom{\binom{4}{1}^5}\binom{13}{5}-10\right]\left[\binom{4}{1}^5-4\right]=1302540\]It is necessary to select ranks in such a way that no multiples of the same rank occurs, but it's also necessary to ensure that the hand is not a straight or a flush.
First, determine the combinations of 5 distinct ranks out of the 13. 10 of these combinations form a straight, so subtract those combinations. Then, select a suit for each of those 5 ranks. This can be done in \(\binom{4}{1}^5\) ways, but 4 of those ways give a flush, so subtract those ways. Using the rule of product, multiply the number of ways to select the ranks by the number of ways to select the suits:
\[P(\text{High Card Hand})=\frac{1302540}{2598960}=\frac{1277}{2548}.\ _\square\]
Probability of One Pair Hand
\[P(\text{One Pair Hand})=\frac{352}{833}\approx 0.422569\]\[\text{One Pair Hand Frequency}=\binom{13}{1}\binom{4}{2}\binom{12}{3}\binom{4}{1}^3=1098240\]First select 1 rank out of the 13 for the pair. Then, select 2 suits out of the 4 for the pair. Then, select 3 distinct ranks from the remaining 12. Then, select a suit for each of those cards. As all of these selections are independent, the rule of product can be used to calculate the total frequency:
\[P(\text{One Pair Hand})=\frac{1098240}{2598960}=\frac{352}{833}.\ _\square\]
Probability of Two Pair Hand
\[P(\text{Two Pair Hand})=\frac{198}{4165}\approx 0.047539\]\[\text{Two Pair Hand Frequency}=\binom{13}{2}\binom{4}{2}^2\binom{11}{1}\binom{4}{1}=123552\]First, select 2 distinct ranks out of the 13 for the two pairs. Then, select 2 distinct suits out of the 4 for each of those pairs. Then, select a rank (out of the 11 remaining) and a suit for the final card. As all of these selections are independent, the rule of product can be used to calculate the total frequency:
\[P(\text{Two Pair Hand})=\frac{123552}{2598960}=\frac{198}{4165}.\ _\square\]
Probability of Three of a Kind Hand
\[P(\text{Three of a Kind Hand})=\frac{88}{4165}\approx 0.021128\]\[\text{Three of a Kind Hand Frequency}=\binom{13}{1}\binom{4}{3}\binom{12}{2}\binom{4}{1}^2=54912\]First, select a rank for the three cards of the same rank. Then, select 3 suits out of the 4 for those cards. Then, select 2 distinct ranks out of the remaining 12 for the last two cards. Then, select a suit for each of those cards. As all of these selections are independent, the rule of product can be used to calculate the total frequency:
\[P(\text{Three of a Kind Hand})=\frac{54912}{2598960}=\frac{88}{4165}.\ _\square\]
Probability of Straight Hand
\[P(\text{Straight Hand})=\frac{5}{1274}\approx 0.003925\]\[\text{Straight Hand Frequency}=\binom{10}{1}\left(\binom{4}{1}^5-4\right)=10200\]A straight can begin with any rank between A and 10; thus there are 10 possible ways to choose the ranks for a straight. Choose 1 of these ways. Then, choose a suit for each of those cards. However, 4 of those ways to choose suits are flushes, so subtract 4 from that amount. Multiply the number of ways to choose the ranks by the number of ways to choose the suits to obtain the total frequency:
\[P(\text{Straight Hand})=\frac{10200}{2598960}=\frac{5}{1274}.\ _\square\]
Probability of Flush Hand
\[P(\text{Flush Hand})=\frac{1277}{649740}\approx 0.001965\]\[\text{Flush Hand Frequency}=\left(\binom{13}{5}-10\right)\binom{4}{1}=5108\]First, select 5 distinct ranks out of the 13. However, 10 of those combinations are straights, so subtract 10 from the number of ways to select ranks. Then, select a suit. Multiply the number of ways to select ranks by the number of ways to select suits to obtain the total frequency:
\[P(\text{Flush Hand})=\frac{5108}{2598960}=\frac{1277}{649740}.\ _\square\]
Probability of Full House Hand
\[P(\text{Full House Hand})=\frac{6}{4165}\approx 0.001441\]\[\text{Full House Hand Frequency}=\binom{13}{1}\binom{4}{3}\binom{12}{1}\binom{4}{2}=3744\]First, select a rank for the three-of-a-kind. Then, select 3 suits for those cards out of the 4. Then, select a rank from the remaining 12 for the pair. Then, select 2 suits for those cards. As all of these selections are independent, use the rule of product to find the total frequency:
\[P(\text{Full House Hand})=\frac{3744}{2598960}=\frac{6}{4165}.\ _\square\]
Probability of Four of a Kind Hand
\[P(\text{Four of a Kind Hand})=\frac{1}{4165}\approx 0.000240\]\[\text{Four of a Kind Hand Frequency}=\binom{13}{1}\binom{4}{4}\binom{12}{1}\binom{4}{1}=624\]First, select a rank for the four-of-a-kind. Select all 4 suits for those cards. Then select a rank (out of the remaining 12) and a suit for the final card in the hand. As all of these selections are independent, use the rule of product to find the total frequency:
\[P(\text{Four of a Kind Hand})=\frac{624}{2598960}=\frac{1}{4165}.\ _\square\]
Probability of Straight Flush Hand
\[P(\text{Straight Flush Hand})=\frac{3}{216580}\approx 0.000014\]\[\text{Straight Flush Hand Frequency}=\binom{10}{1}\binom{4}{1}-4=36\]Select 1 of the 10 possible combinations of ranks that gives a straight, then select a single suit for all 5 cards. This gives the number of straight flushes, but 4 of those hands are royal flushes, so subtract 4 from that amount:
\[P(\text{Straight Flush Hand})=\frac{36}{2598960}=\frac{3}{216580}.\ _\square\]
Probability of Royal Flush Hand
\[P(\text{Royal Flush Hand})=\frac{1}{649740}\approx 0.000002\]\[\text{Royal Flush Hand Frequency}=4\]This one is easy! There is only one kind of straight that can make a royal flush, and it can be any of the 4 suits. Thus, there are only 4 possible royal flushes:
\[P(\text{Royal Flush Hand})=\frac{4}{2598960}=\frac{1}{649740}.\ _\square\]
Each of these probabilities assumes that you are only dealt 5 cards. In an actual game of poker, the manner in which cards are dealt can vary, and this will affect the probability of each classification of hand.
You are playing a game of poker, and you are dealt the following hand of cards from a shuffled standard poker deck:
\[\text{A}\spadesuit, \text{A}\clubsuit, \text{A}\color{red}{\heartsuit}, 6\color{red}{\heartsuit}, 10{\color{red}{\diamondsuit}}.\]
You put the \(6\color{red}{\heartsuit}\) and \(10\color{red}{\diamondsuit}\) cards aside, and request to be dealt two new cards.
What is the probability that you will improve your hand to a Four of a Kind or a Full House?
Round your answer to three decimal places.
Note: Cards dealt to you are no longer in the deck. The \(6\color{red}{\heartsuit}\) and \(10\color{red}{\diamondsuit}\) cards are put aside; they are not put back into the deck.
You and a friend are playing poker together. After soundly defeating your friend for several rounds in a row, you offer your friend the following handicap:
You will play with part of a standard poker deck consisting of only the cards 2 through 6 (20 cards), while your friend will play with the remaining cards (32 cards). You will play a game of poker in which each player is dealt 5 cards and there is no 'discard and replace' phase. The normal rules for poker hand superiority apply.
If the probability that you win a round of this version of poker is \(P\), then what is \(\lfloor1000P\rfloor ?\)
Common Rules of Poker
Each variant of poker tends to have the following features in common.
Buy-in
Poker typically requires that players put down money before they play the game. This is called a buy-in. The buy-ins are a prize given to the winner. The purpose of buy-ins is to ensure that each player has a stake in playing well and winning the game.
Betting chips are used to represent money while playing. Sometimes, players are allowed to put down more money in the middle of a game, but players are typically not allowed to "cash out" their chips until the game is over.
Dealing
Each round of poker has a dealer. This person is responsible for shuffling the deck and dealing the cards to each player. Sometimes, a non-player is given dealer responsibilities for the entire game. Otherwise, each player takes turns being the dealer. A dealer chip is used to designate who is the dealer each round, and that chip is passed on to a new player after each round. Even if the dealer is a not a player, this chip is still passed around, because certain betting rules depend of the location of the dealer at the table.
Betting
The pot: The total amount of money bet by players each round is called the pot. The winner of each round takes the entire contents of the pot for that round. If there is a draw after a round, then the pot is shared among those players in a draw.
Ante: Many variations of poker require each player to bet a certain amount before each round begins. This is called an ante bet. The ante happens before players see their cards. The purpose of this rule is to prevent games from going on too long, and to keep each player somewhat invested in each round.
Blinds: Some variations of poker require blind bets. These bets can replace the ante, or they can be in addition to the ante. Like an ante, they happen before each player is dealt their cards. Unlike an ante, only some of the players are required to make a blind bet. This requirement is rotated around the table each round so that each player takes turns making the blind bet.
Betting: The main betting phase typically begins after players have been dealt their cards. Betting begins with a different player depending on the variant of poker. Each player takes a turn betting, and these turns are taken clockwise around the table. There are a number of options each time a player takes his or her turn betting:
Check: If no money was raised since the player's last turn, that player can check and pass to the next player. If the round has a blind bet, then each player must call the blind bet before they can check.
Call: If money was raised since the player's last turn, that player can call and bet money equal to the difference in the amount of the current bet and the amount that the player last bet.
Raise: A player can raise the amount of the bet by betting more money than the current bet.
Fold: A player can refuse to bet. This is called a fold, and that player is effectively out of the round. A player that folds gives up all money that he or she bet that round. It may seem wasteful to fold, but this is often the best strategy when a player knows that he or she is not likely to win the round.
All-in: In certain situations, a player will put all of his or her remaining chips into the pot. This is called an all-in. There are special rules for how this type of bet works, depending on the variant of poker.
The round of betting is over once each player at the table has either called, checked, folded, or made an all-in bet.
Winning a round
For each round, there is a final betting phase. The round is over after this betting phase. Only the players who have not folded have a chance to win the round. Players take turns clockwise around the table revealing their hands. The player that begins this process depends on the variant of poker. A player may choose not to reveal his or her hand, but a player who makes this choice cannot win the round.
The player that wins the round is the player with the best 5-card hand. This player wins all the money in the pot. Sometimes, there is a tie among the best 5-card hands. In this case, the round ends in a draw, and the pot is shared among the players with those hands.
Winning the game
Over the course of many rounds, players will run out of money and drop out of the game. The game is over when one player has won all the money that was put down as buy-in at the table.
Even though the winner has won all of the chips at the table, there are often rules for how this money is shared after the game is over. It can be agreed before the game starts that the last remaining players will share the money in some way. This ensures that the game is not all-or-nothing; players can win some amount of money if they play well, even if they don't win the game.
Variants of Poker
It's important to know the rules of a poker game to be able to calculate probabilities in poker. There are many variants of poker; the following are a couple of the most common:
Five-Card Draw
This is regarded as the simplest version of poker to learn.
For each round, ante and/or blind bets are made. After the ante and blinds, each player is dealt a hand of 5 cards. Players look at their cards, and keep them hidden from other players. The first betting phase begins after each player has seen his or her cards. Betting typically begins with the player to the left of the dealer or to the left of the player with the blind bet.
The next phase of the round is called the draw phase. During this phase, players can choose to discard cards from his or her hand and request to be dealt that many cards. Players will typically use this phase to improve their hands to more valuable hands. In some versions of five-card draw, there is a limit on how many cards can be discarded and replaced. However, most of the time, there is no limit on the number of cards that can be discarded and replaced. A player could discard his or her whole hand for a new hand if that player wished.
After the draw phase, the final betting phase begins. Afterwards, players take turns revealing their cards. Whoever has the best hand wins the pot.
Then, a new round with antes and blinds begins.
Seven-Card Stud
This variant of poker is a stud, meaning that each player has some cards that are revealed to all players at the table. Each player is dealt a total of 7 cards, but each player's hand is only the best 5-card hand out of those cards. Other than the first 3 cards, players are dealt cards one at a time, with a betting round between each newly dealt card.
For each round, ante and/or blind bets are made. After the ante and blinds, players are dealt 2 cards face-down (hidden from other players) and 1 card face-up (revealed to other players). The first round of betting begins either with the player who has the best face-up card, or with the player to the left of the player who blind bets.
After the first betting phase, each player is dealt a card face-up. Then, another betting phase begins with the player who has the best face-up cards.
After the second betting phase, each player is dealt a card face-up. Then, another betting phase begins with the player who has the best face-up cards.
After the third betting phase, each player is dealt a card face-up. Then, another betting phase begins with the player who has the best face-up cards.
After the fourth betting phase, each player is dealt a card face-down. Then, the final betting phase begins with the player who has the best face-up cards.
After the final betting phase, players make the best 5-card hand out of their 7 cards. Players take turns revealing their cards, and the player with the best hand wins the pot.
The structure of each phase can be summarized as follows: 2 down and 1 up, bet, 1 up, bet, 1 up, bet, 1 up, bet, 1 down, bet.
After the round is over, a new round with antes and blinds begins.
Texas Hold-Em
This is now the most popular variant of poker. It is a variant of community card poker: In this kind of poker, some cards are revealed to the whole table, and each player can use those cards to build his or her 5-card hand.
A round begins with blind bets, and sometimes ante bets. Texas Hold-em typically has a "big blind" and a "small blind." The big blind is an amount twice as much as the small blind. The player to the left of the dealer makes the small blind bet, and the next player to the left makes the big blind bet.
After these bets, each player is dealt 2 cards face-down (hidden from other players). This phase is called the pre-flop, and each player's hidden cards are called that player's hole or pocket. The first phase of betting begins with the player to the left of the big blind.
After the pre-flop betting phase, 3 cards are dealt face-up (revealed to all players) at the center of the table. These 3 cards are called the flop. They are community cards, meaning that each player uses them to build his or her 5-card hand. After the flop is dealt, another betting phase begins with the player to the left of the dealer.
After the flop betting phase, another community card is dealt face-up next to the flop. This card is called the turn. After the the turn is dealt, another betting phase begins with the player to the left of the dealer.
After the turn betting phase, another community card is dealt face-up next to the others. This card is called the river. After the river is dealt, a final betting phase begins with the player to the left of the dealer.
Each player still in the round reveals their hands simultaneously. Each player makes the best possible 5-card hand available from his or her pocket cards and the community cards. Because Texas Hold-Em uses community cards, ties are more common than with other variants, and special rules designate how to break ties based on the specific cards contained in each player's hand. Even so, draws are still possible, and the pot is shared if this is the case. Otherwise, the player with the best 5-card hand wins the pot.
After the round is over, a new round with antes and blinds begins.