Texas hold 'em and Omaha are two well-known variants of the community card family. There are several methods for defining the structure of betting during a hand of poker. The three most common structures are known as 'fixed-limit', 'pot-limit', and 'no-limit'. In fixed-limit poker, betting and raising must be done by standardized amounts.

Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold'em is the version played most often at casinos and is the most popular among the 'community cards' variants of poker. It is also the variant played at the World Series of Poker and on the World Poker Tour.

Rules

Each hand is played with a whole deck. One player is the dealer (this role rotates clockwise after each hand), and bets are placed in a clockwise order starting with the player on the dealer's left. Each hand has four stages, and after each stage there is a round of betting. The four stages are:

1. (Pre-flop) Everyone gets two cards (dealt face down).
2. (Flop) Three cards are dealt face up in the middle of the table.
3. (Turn) A fourth card is dealt face up in the middle of the table.
4. (River) A fifth card is dealt face up in the middle of the table.

The object of the game is to form the best five-card hand possible using the player's two cards and the five 'community cards' dealt in the middle of the table. A hand is won by having the best hand among the players who did not fold (i.e. refuse to match an opponent's bet, as described below), or by having everyone else fold.

We are going to use a $1/$2 betting structure. Before the pre-flop, the two players to the left of the dealer must bet $1 (these mandatory bets are called blinds, since the player must make them before she sees her cards). Then, following the betting order, each player may raise the bet, up to four times per player per betting round. (The blinds act as a bet, so in the pre-flop betting round, the first player to act will be the person three seats to the left of the dealer). Whenever a player raises the bet, the other players must call (that is, accept the raise), fold (that is, give up and lose the money already bet) or raise the bet even more. On the pre-flop and flop, theplayers bet $1 at a time, while on the turn and river they bet $2 at a time.

The hand ends when all but one player has folded or when all the cards have been dealt and the last betting round is over. In this last case, the players must show their cards and the player with the highest hand wins.

Ranking of Poker Hands

From highest to lowest, the possible five card hands in poker are ranked as follows:

• Straight Flush Five cards in sequence all of the same suit, for example 7 - 8 - 9 - 10 - J all in hearts. Aces can be treated as either above kings or below twos. If multiple players both have straight flushes, the players compare the high cards of their straight flushes (higher card wins). A straight flush with A - K - Q - J - 10 is called a royal flush and is the highest hand in the game.
• Four of a Kind Four cards of the same rank (and one card of some other rank). When comparing four of a kind hands the four of a kind of higher rank wins.
• Full House Three matching cards of one rank and two matching cards of a different rank. If multiple players both have full houses the player whose set of three cards has higher rank wins.
• Flush All five cards of the same suit. If multiple players both have flushes, the player with the single card of highest rank wins. In situations where players both have the same rank high card they then compare the rank of their second highest cards, then third highest cards, etc. Two flushes with cards of the same ranks tie.
• Straight All five cards in sequence. As with straight flushes, aces can be high or low and two straights are compared by looking at high cards.
• Three of a Kind Three cards of the same rank and two unmatched cards. Three of a kind hands are compared by looking at the rank of the set of three cards.
• Two Pair Two cards of the same rank, two cards of another rank (matching each other but not the first two cards), and one card with a third rank. Two of a kind hands are compared by looking at each player's highest ranking pair (higher ranks wins). If players have the same high pair then they compare the rank of their low pairs. If this still does not decide a winner then the rank of the unmatched card (also called the 'kicker') is compared.
• One Pair Two cards of the same rank and three cards all of different ranks (both from the pair and from each other). One pair hands are compared by looking first at the rank of each player's pair, and if needed then considering the rank of each player's unmatched cards in the same way as for a flush.
• High Card A set of five cards that does not fit into any of the above categories. High cards hands are compared by considering the rank of each player's cards as described above in the case of a flush.

There is no ranking of suits in poker, so two players who have identical hands but in different suits tie the hand and split the pot. The two cards that a player does not use in making his five card hand are ignored; they are not used to break ties between five card hands.

Some Calculations

Pot odds are the odds you get when you analyze the current size of the pot against the cost of your next call. The general idea is to compare your chance of winning to your pot odds. You have good pot odds if your chance of winning is significantly bigger than the ratio of the bet to the pot size. For example, say you are on the turn, you have two hearts in your hand, and you have one opponent still in the hand. The community cards have two hearts, so any of the nine remaining hearts finishes a flush for you. We say that you have 9 'outs' (outs are the cards still unseen that will improve your hand) out of a total of 46 unseen cards. The ratio 9/46 is approximately 1/5. Suppose your opponent raises $2 and the pot you get if you call and win is $20. The ratio 2/20 is 1 in 10, which is smaller than your 1 in 5 chance of hitting the flush, so pot odds say that calling is the right move.

Implied odds take into account the fact that betting will continue throughout the rest of the hand, so you have the potential to gain more money from your opponents in future rounds of betting (and also you may have to pay more money to stay in the hand in later rounds of betting). In the example above, if you feel that your opponent will call a bet after the river, then if hit your flush you will be able to earn an additional $2. If you do not hit your flush you can fold the hand and not lose any additional money. So in this case your implied odds are 2/22, or 1 in 11, even better than your pot odds.

For all of the following problems we assume that all of the cards not in a player's hand or in the collection of community cards are drawn with equal probability. This is a valid assumption if we have no knowledge of the other players' cards (see the Blackjack lesson, problem 3 for further details). In actuality it may be possible to infer some information about an opponent's hand based on her betting patterns or behavior.

1. If you are dealt two hearts and the flop contains exactly two hearts, what is the probability that you get a flush on the turn or the river? If the flop contains only one heart, what is the probability that you get hearts on both the the turn and the river to make your flush?
2. You are dealt a pair of eights and the flop comes up 1 - 7 - 2. What the is probability that you will have four a kind after the river? A full house?
3. Your are dealt a 6 - 4 and the cards on the table are 7 - K - 3 - 10. There are two opponents still in the game. The pot is currently $20 and your have been raised $2. Assuming that you win if you hit your straight and lose if you do not, what do pot odds tell you to do? Assuming further that both of your opponents will call a $2 bet after the river, what do implied odds tell you to do?

4. It is sometimes useful to know the frequency of each of the different poker hands. In Texas Hold'em, each player is making a hand out of seven available cards. To determine the probability of each hand occuring we can count the number of distinct ways of obtaining each hand and divide by the total number of possible hands. This requires thinking about all of the different ways of obtaining a given hand and coming up with an orderly process for counting these different ways. The easiest way to do this involves heavy use of combinationsC_n,m (the number of ways of choosing m objects (in any order) from a collection of n objects. These are also called binomial coefficients. See the probability review for more details). For example, the total number of possible seven cards hands is equal to the number of ways of choosing seven distinct cards out of a collection of 52, giving a total of C_52,7 = 133,784,560 hands. Note that even though two identical hands in different suits have the same value in poker they are being counted as distinct hands.

   The difficulty of calculating these frequencies varies significantly by hand. The high ranking hands such as four of a kind and straight flush can only be obtained in a limited number of different ways and therefore it is therefore easier to calculate their frequencies. Try calculating these frequencies first. Once you get the hang these types of calculations, if you are up for a challenge you can attempt some of the more involved calculations.

Solutions

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As a college professor, I am always eager to bring poker examples into my classroom. Since I primarily teach probability and statistics courses, this tends to work well. I was thus very pleasantly surprised to come across a textbook titled, 'Introduction to Probability with Texas Hold’em Examples'. Written by Dr. Frederic Paik Schoenberg, Professor of Statistics at UCLA, this book is intended to be used in a college-level, calculus-based probability course, with every single example in the textbook drawing on the game of Texas Hold’em.

Review of the Textbook

Upon coming across this book, my first thought was, “Someone beat me to it.” But my next thought was, “Can this book actually be any good?” Despite the natural link between poker and probability, the majority of academic statisticians I have met are not really that into poker, with only a few exceptions coming to mind.

However, a quick scan of the book proved my skepticism to be wrong. Very wrong. Dr. Schoenberg’s wealth of examples using real hands from the World Series of Poker, High Stakes Poker, and other major events across the past couple of decades is truly impressive. Additionally, his depth and breadth of poker understanding is made apparent by his ability to place these hands in context and use them to ask meaningful questions from both a poker and a probabilistic standpoint. The usage of real data has been repeatedly championed as a cornerstone for best practices in statistics education, and while it is often easier said than done, Dr. Schoenberg has accomplished just that in a very impressive way throughout his book.

A Need for Real Examples

In Dr. Schoenberg’s own words, he writes in the preface to the book, “The first few times I taught probability, I was disappointed by many of the examples in the books. They typically involved socks in sock drawers or balls in urns. Most of my students did not even know what an urn was, and certainly were not motivated when informed of its meaning. I thought it would be interesting to try to teach the same topics as those covered in most probability texts using only examples from poker.”

So, this coming fall semester, I will be enthusiastically using this textbook as a required course text in my 300-level probability course.

Potential Bias

There are, however, a few concerns that have been brought up by some of my friends and colleagues. Firstly, there is a concern that using poker as a pedagogical platform may introduce a gender preference: that is, as much as we would like this not to be true, the current reality is that the average male tends to be more interested in poker than the average female, and probably more than most people who are anywhere else on the gender spectrum (though my anecdotal sample size is a bit smaller there). Second, the poker background that my students will come in with will likely be highly varied: some students may already be serious poker players, whereas others may have never even played Texas Hold’em before. To the extent that any of this might put any students at a disadvantage as far as their learning and/or their grade is concerned, I would definitely want to be cognizant of it.

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Addressing the Concerns

To the first concern: I do not believe that there is an inherent Y-chromosomal predisposition to the enjoyment of poker. Rather, I believe that the imbalance that we currently see is largely due to societal gender norms; specifically, men have simply been exposed to poker at a much higher rate than women or any other gender. To a large degree, then, I believe that any gender preference induced by using poker in the classroom is, in essence, a specific case of a preference for those with more poker background to begin with. This is undoubtedly an oversimplification, but a deeper discussion is beyond the scope of this article.

Thus, assuming at least some level of accuracy to my above observation, I plan to simultaneously mitigate these concerns by beginning my class from the ground up as far as poker knowledge goes. We will begin on Day 1 by introducing everyone to the rules of Texas Hold’em, and I will aim to be very aware of any students who come in with little to no background in the game. Moreover, in line with a learn-by-doing philosophy that is integral to my teaching style in general, we will spend some class time that day actually playing the game.

I also plan to have a few poker nights for the class throughout the semester that will be “optional but highly recommended” (for free with the possibility of winning prizes that I will provide). Through this, my goal is to foster at least a modest level of curiosity and interest for the game among those who may not already have it, and enhance it among those who do, with the ultimate aim of fully utilizing the power of a fun game as a gateway towards the academic topics of probability that we will cover in the course.

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Follow the Journey

Even among my most gambling-averse students or those who aren’t even into games in general, I believe that at the very least they will find Texas Hold’em to be more interesting than balls in urns. As is true on the first run of anything, though, I am sure that there will be bumps along the way, both anticipated and unforeseen. If you’d like to stay tuned to my experience, I plan to blog about it each week with an open comments section where I welcome any questions or feedback.

Peter Chi is an Assistant Professor of Statistics at Ursinus College, a small liberal arts college outside of Philadelphia, PA. While his professional interests lie mainly in biomedical research and evolutionary biology, he is an avid recreational poker player and enjoys using poker and other gambling examples in the classroom. Follow him on Twitter at @PeterBChi.

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