The Mathematics of Poker

  

A variety of variables, such as thrill, monetary value, and most importantly the skill, have all contributed to the demotic nature of poker today. The popularity of poker has taken off incredibly since 2003, when many whom hadn’t even heard of the game of Texas Hold’Em, tuned in to the World Series of Poker to watch Chris Moneymaker pull of incredible plays and bluffs to take down the most coveted prize in poker. A pure amateur, at the time, managed to take home the WSOP Main Event bracelet and a $2.5 million payday.

Since then, and especially after watching Moneymaker’s historic run to the title, many players have debated the necessary skill involved in making it to the top. Whether it be winning a tournament satellite, raking in a pot at your local cash game, or being successful on one of poker’s grandest stages, most astute players of the game recognize that it often requires a bit of luck.

Although luck is undoubtedly an element of playing poker, it’s essential to be aware of the fact that everyone’s (yes, I said everyone’s) luck is going to be the same. The separation of skill and expertise that you share over your opponents however, can be very diverse. This difference, is what’s most important. While nearly every professional player will tell you that knowing your opponent’s tendencies, tells, and poker background will increase your edge over the long run, it’s just as integral to know and understand the mathematics and probability of the game of poker. Even though the prize pool payouts are numbers you shouldn’t forget, they certainly aren’t the only statistics you should be keeping track of.

Going back to 2003, Sammy Farha played arguably one of the biggest poker hands of the century against Chris Moneymaker in the World Series of Poker Main Event. (Feel free to take a look here.) Sammy Farha is a professional poker player who’s been playing for several decades. Farha also has a ton of experience playing heads-up poker. So you would assume, that with a substantial edge in experience, knowledge, and skill, how would an amateur from Atlanta, Georgia end up winning the WSOP bracelet instead of him?

Well, at first, we might say that’s about as probable as me beating Michael Jordan in a game of H.O.R.S.E. But we haven’t yet considered all of the factors. Let’s look at some mathematical elements. Beginning heads-up, Farha was at a chip count of 3.7 million, to Moneymaker’s 4.6 million. Although not a huge edge, advantage Moneymaker. Despite having seen Moneymaker play a few hundred hands at the Main Event final table, most players (including Farha) would consider it very difficult to get an accurate depiction of a player’s playing style through such a small sample size. Farha almost certainly considered the same idea. With the lack of a true read, Moneymaker now holds another edge on his adversary, because of a lack of information. Once again, advantage Moneymaker.

If we consider all of that information, we can start to dig even deeper into the mathematics and minds of both players in that particular hand. On a board of 9s2d6s, while Farha’s flop check was rather standard, the lack of a continuation bet by his opponent who raised pre-flop was anything but. Moneymaker had been playing aggressively throughout most of the tournament, and although Farha knows his opponent is a rookie, he also recognized that Moneymaker was capable of making above average plays against his opponents. However, Farha may have recognized just a moment too late. But let’s stop for a second.

It’s important to mention, that Moneymaker hadn’t been making many, if any, bluffs at all during the latter stages of the Main Event. It’s improbable as well, for an amateur player to be bluffing against one of the best professional players in the world at that time. In essence, the poker world had never seen such a play made by a player of Moneymaker’s skill level, against such a worthy opponent. It was mind boggling, and in many ways, it still is.

With that said, it’s understandable for Farha to fold his pair of 9s in that particular spot, because during that time a large percentage of his professional peers would’ve agreed it was the correct (and most probable) decision to make. But we quickly learned, that Moneymaker wasn’t like most amateurs, and unfortunately, that was nearly impossible for Farha to predict. So Farha made the smart (although wrong) choice to rely on probability.

Although playing H.O.R.S.E straight up against Jordan would be difficult, the game changes dramatically if he has to start with an H and an O. Add in the fact that he has no idea that you played high school basketball, and you can shoot a pretty decent 3-pointer, and well, you may just have a shot. This is a reasonable analogy to the hidden mathematical side of poker.

Poker is a game in which as players we must use all of the information that’s available to us. Whether it be a player’s stack size, the amount of money in a pot, or how likely we are to make a flush, we must analyze and apply that knowledge to increase our win rate at the table.

Probability

One of the most used phrases when it comes to poker math, knowing the probability of some frequent poker situations can be extremely helpful to your poker career. Often when we play poker, we play to the strength of our cards. Meaning; often times when we have AK, we play them actively and aggressively. Or, when we hold 33, we call casually, hoping to flop a set. Now wouldn’t it be useful to know how often we’ll flop a set when we’re holding a pocket pair? Or how often we’re going to connect on the flop when we’re dealt AK? You don’t have to answer that. I wouldn’t be writing this article if the answer was no.

Although I could go on all day calculating the probability of some common poker situations, it’s more important to make you aware that most poker occurances CAN be calculated. Knowing just how likely it is that your opponent flopped a set against you, is going to make your decisions a little easier when you’re holding a big pocket pair. It might be essential to know how frequently you’ll catch your flush draw on the turn, since you decided to play back against a tight-aggressive regular. Staying acute on many of the ordinary happenings at the felt will put you ahead of many of your adversaries, and may even inspire you to dig up that statistics textbook your old college bookstore refused to take back.

(By the way, the odds of flopping a set are about 11.8%, or approximately 1 in every 8 tries. As far as flopping an ace goes, along with any other unmatched hole card, that’s about a 32% chance, or 2 to 1 against.)

Analyzing Odds

Although a close second to probability, calculating your odds at the poker table is a very important skill. Determining whether or not your getting the right price to make a call can be critical to your long-term win rate.

When determining your odds at the poker table, we need to distinguish between two very important characteristics.

Pot odds, and implied odds.

Let’s start with the former. Pot Odds, in Layman’s terms, is the monetary ratio you’re given to make a call on your opponent’s bet. In terms of mathematical formulas, it’s the total amount of the pot (including your opponent’s bet), divided by the amount you would need to make the call. Below, I’ve included a visual representation of the formula, although sometimes, it’s best just to give an example.

(Total amount of pot) / (Your call) = Pot Odds

Let’s say there was a raise pre-flop to $11 from early position in your local $1/$2 No-Limit Hold’Em game. We made the call from the button. Both blinds folded. In the pot (not considering rake) there’s now $25. ($11 x 2 = $22, plus both blinds.) Your opponent takes the lead on the flop, and bets $10. Now, with that information, let’s calculate your pot odds.

There’s $25 already in the pot, plus the addition of your opponents flop bet. So now, there’s a total of $35. In order to make the call, you need to add $10 of your own. With that information, we now have all that we need. Using the formula stated above, this gives us the ratio of:

$35 / $10 or 3.5 / 1.

To make the ratio as simplistic as possible, we always divide the denominator by itself, in order to get a denominator of 1. And as most know, what you do to one side of a ratio, you must do to the other, so we divide 35 by 10, to get 3.5. Got it? Good. (On a side note, if you’re ever given 3.5 to 1 odds or better, it’s almost safe to call any bet with your draw. But use that information carefully!)

Although I’m positive that most of you who are reading this can add, subtract and divide, even I will admit that during a poker hand, calculating your pot odds can be difficult. As a word of advice, I always recommend adding things up as you go. It’s much easier to figure out things in steps, rather than squint into one huge mush of chips pooled in the center of the table, hoping to find the hidden meaning of pot odds. Even if you are great at math, it presents much more room for error if you try to calculate your amounts all at once. But enough with that gibberish. Here’s where it gets good.

The intricate formula I just presented, means absolutely nothing. Yes, nothing. UNLESS… it’s coupled with something we like to call “equity“. Equity, we will explain later in this article. So we’ll come back to this in a bit.

Although sometimes calculating pot odds can be challenging, the world of implied odds is in many ways the grandmaster of poker concepts. Despite being complex, it’s also very important to understand. So don’t be afraid! I’ll attempt to explain implied odds as best as I can.

Implied odds are the odds you’ll be getting on later streets (turn, river, etc.) in consideration of what the bets will most likely be. Many also define implied odds as; the chances of you winning your opponent’s entire chip stack, if you happen to make your hand. While this sounds somewhat simple, unlike pot odds, there isn’t a mathematical formula for calculating this. Instead, what players must do, is closely analyze the information they’ve been given about their opponents in order to make an informed decision. In example; Does your adversary fold to any scare card? Is she extremely aggressive with her “made” hands? Is she a “calling station?”

This information is dire because you need to understand how your opponent might react once your draw has completed. If you’re playing against Mr. Tighty (for lack of a better phrase), you need to recognize that he’s probably not going to pay you if you make a large river bet with a third spade on board. It may be best that you fold your draw in that position, because even if you hit, it’s unlikely you’ll make any more money. On the contrary, if your immediate odds are only laying you a measely 1.5 to 1, but you just happen to be in a hand against Mr. Call-y, then you should be aware that your implied odds recommend that you call and get paid by him on the river. (Just in case you didn’t know, Mr. Call-y calls most bets.)

The ability to study your opponent’s moves, coupled with your ability to calculate and recognize implied odds will make your overall poker game sore through the roof. But I can’t stress enough, you must carefully dissect the appropriate situations and times in which to use implied odds. In a limped pot, when there’s been minimal action, it would be foolish to expect to be paid gracefully with the absolute “nuts.” Remember, you must be conscious of the fact that you must be offered a strong chance at winning your opponent’s stack to make a call reasonable.

(Interestingly enough, the article I’ve recently written on ranges will help considerably, when analyzing what hands your opponents might be playing, and at what times they just may be willing to pay you off on your monster hand. Take a look.)

Equity

Time for Part Two! This is the bread-and-butter of this strategy article. Well sort of. Equity can be defined as the percentage chance that you have at winning a hand. In other words, how often will you actually make your hand. In order to figure out what you’re equity is, you need to be able to count the amount of “outs,” or cards that you can catch to make your hand.

Before we do that however, we must take note of some obvious, but important information.

*In a 52-card deck, there are 13 cards of each suit, and there’s 4 cards of each value.

So in an example, let’s say you have a flush draw. There are two hearts in your hand, plus two hearts on the board. This means there are only 9 hearts left, out of 13. So, you have 9 outs left to make your hand.

*When calculating equity, there’s a very simple mathematical formula. When estimating your equity on the flop, you multiply your outs by four. When doing so on the turn, you multiply your outs by two.

Now as important as that is to remember, I also must state, that when using this shortcut, it doesn’t ALWAYS produce an accurate answer. I know, I know. Well why would I tell it to you then?! Don’t pull a hissy fit just yet. Most of the time, it does. But when calculating equity with a large number of outs, specifically anything greater than 8, the above formula produces a slightly incorrect answer. You’ll need to use a different formula. This is given below.

Equity = {# of Outs x 4} – {# of Outs – 8}

This formula gives us a much more accurate solution when it comes to equity. So using a few normal situations, if you had that flush draw on the flop, your equity percentage would be:

{9 x 4} – {9 – 8} = 35% EQUITY

Now, although this particular example would only yield a 1 percent change when considering the two formulas, if you calculated equity with 15 outs, there would be a large difference between having 60% equity, and 53%. Having the most accurate information possible will allow you to make the most accurate decision when choosing whether or not you’re “getting the right price” to call a bet with your draw. <– This, is essentially the most important part of calculating your pot odds and equity, and also, this article.

Before we get into that comparison, it’s critical to note the idea of anti-outs. Anti-outs are simply outs that help you make your hand, but also complete a better hand for your opponent. So in example, if you happen to make your straight on the turn, but your opponent makes a flush, this would be an example of an anti-out. When determining how many outs you REALLY have to make a winning hand, it’s important to be able to estimate your opponent’s holdings accurately, so that you can determine if any of your outs are indeed a dud. If you believe one of your outs may complete a more dangerous hand for them, don’t include it in your calculations. When in doubt, it’s always best to assume your opponent has the hand that’s the most detrimental to your own, so that you can appropriately adjust your equity.

To compare them, we need to make sure both our odds and equity are in the same format. Since I’m keen on fractions, and I believe they’re easier to work with than percentages, let’s convert our equity to a ratio.

Let’s say we’re getting pot odds of 2.5 to 1, or 2.5/1. You’re holding a flush draw on the flop. We figured out earlier, that your equity in this instance is 35%. (For simplicity, we’re going to assume all of your outs are available.) Now in order to convert this percentage into a fraction, we’re going to need to do a little math. We know, that ultimately 35% can be considered 35 out of 100. This also means, that your opponent has a 65% chance (65 out of 100) to beat you. But since we’re looking for our odds, or the “against” ratio, we need to realize that our fraction will be 65/35 against. We’re behind in the hand in terms of equity, so the bigger number goes on top. After doing a little seventh-grade improper fraction division, we determine that our ratio, or equity, is 1.8 to 1 against. This means, that in order for us to make money over the long run, our pot odds need to be greater than 1.8 to 1. Fortunately enough for us, THEY ARE!!

So make that call!

After diving head first into a plethora of poker mathematics, it’s now time to put it all together. How exactly is all of this information important? Pot odds and equity, when combined allow poker players to make correct long-term decisions, in terms of profitability at the poker table. Taking a page out of a college textbook in logical reasoning, when both the monetary value (pot odds) and the statistical probability (equity) are equal, or the former exceeds the latter, it’s always going to make sense to call your opponent, for the sake of long-term profitability and results. While we can be certain that you’re not always going to make your draw by the river, if the pot odds and equity suggest you should do so, your win rate will increase appropriately over time if you continue to make the correct decisions.

Calculating your odds and equity at the table takes lots of practice, and a good amount of memorization. But despite its mathematical characteristics, figuring out when to make the right call at the right time will be an immense step in evaluating your skill level at the poker table. While even I, once an amateur at the game, believed that most of your poker decisions can be made solely by feel, even those who truly believe that notion are often indirectly playing probabilities. Despite them not knowing the exact calculations, those who are ignorant of poker math still chase their draws on the odds of them hitting that miracle card. So why not be completely aware of your chances of doing so?

Aside from holding the “nuts” on the river, there aren’t many decisions in poker that you’ll face where you don’t need to consider your chances of winning. Despite there being tons of other valuable information to gather in your poker sessions, don’t fail to neglect the mathematical importance of poker.

Good luck.

6 Responses to “The Mathematics of Poker”

  1. Matt Alden

    Very informative article. Very approachable for people like me that aren’t good at math. Have a question about calculating equity. How did you get to 35% in your example? I’m guessing its as follows but I wanted to be sure. Thanks!

    ({9 x 4}

    Reply
  2. Kevin

    Good article on the math of hold’em. Very easy to follow. Any suggestions on how to keep a pot count as the hand is happening? Alot of times in a game there are a 100 things going on around you. Do you count as every bet goes in? Or look into the pot and count up that way?

    Reply

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