Master No-Limit Hold'em Cash Games - Join the Elite at GTO Warrior

Beginner's Poker Course Part 2: What Is Expected Value (EV)?

A quick note: this article delves extensively into the concept of Expected Value (EV). If you are already familiar with this concept, you may want to skim or skip this article and jump to the next one, where we will get more practical and start talking about preflop play.

In the first article, we got up in the air a bit and talked about the Universe (or God) telling us the best decisions to make and assigning numbers to each option we can take while playing the game. In this article, we aim to make things a bit more practical and down to earth, taking along that initial perspective of choosing an action with the 'highest value' assigned to it.

Expected Value (EV): The Formal Definition

Let's start by introducing a modern-day term for the numbers we would ideally want to assign correctly to each decision-making option. It's called Expected Value (EV), a well-known concept from probability theory and a fundamental part of poker strategy.

In the first article, we talked about a 'higher power' attaching numbers to each option in poker that would allow you to cheat; those numbers are the expected values of those decisions. If these values were displayed on all the buttons you can click in online poker, and they were correct, well, it would become very easy: simply pick the highest ones every time, and you would win—assuming your opponents didn't pick the highest ones every time.

But how would you pick the 'highest ones' without any help? Well, you could gain an intuitive feel for it through experience, study, and perhaps even some in-game calculations in your head if you can. It's all about choosing actions that have the highest Expected Value (EV) and reaching those conclusions using any means our brains can handle.

How should you think about Expected Value (EV), then? The actual definition, as given by Wikipedia, for instance, is like this:

'In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of the possible values a random variable can take, weighted by the probability of those outcomes.'

Expected Value (EV): A Real-World Example

I know, I know, I don't like reading that wordy Wikipedia definition either. Even for me, as an experienced poker veteran—I mean, yes, I understand what it's trying to convey there, but I would also prefer to read it in simpler language. Let me try to do just that for you.

Perhaps the way I would explain EV to a total beginner, and someone unfamiliar with probability theory, would be with some sort of real-world example. The easiest one I can think of right now is probably this:

Let's say you give me 1 dollar every day, without fail. My Expected Value (EV) for each day is simply 1 dollar.

Now, let's say that occasionally, you decide not to give me that dollar. Perhaps, sometimes, you have a bad day and choose to give it to someone else, or keep it for yourself. Let's assume this happens one day each week, on average…

We're going to assume that I don't know which day it will be; well, my Expected Value now becomes 6/7 of a dollar because there are 6 days out of the week when I receive the dollar, 6 out of 7 days.

My EV is approximately 0.86 dollars, rather than the $1 EV I would have if you'd given me the dollar every day without fail.

Expected Value (EV): A Poker Example

That may be a bit confusing to you as well. Let's get back to the realm of poker. After all, we need to learn how to estimate the EVs of poker decisions.

Let's say we're on the river and we're heads up with our opponent. We have a royal flush; there is no way we can lose.

The pot is $80, and let's say that we both have $100 left in our stacks.

It's our turn to act. We can fold, check, or bet (any amount). The choice with the highest expected value (EV) is quite clear among these options. Naturally, we should bet since we undoubtedly have the best hand; that's the intuitive move. However, what remains less clear—something you'll learn more about as you progress in your poker journey—is how much to bet.

But first, we're going to take a closer look at all the decisions and their respective EVs for this situation, right now:

Folding

It makes intuitive sense that folding is incorrect, and its EV is actually 0 in this specific situation, holding the Royal Flush. That is because we throw away our hand without putting extra money into the pot. We don't lose anything, and we cannot win anything (because we throw our hand away!).

We actually do not lose money by folding, because the money that we put in earlier in the hand, in previous betting rounds, was lost during those decision rounds; EV-wise, the money was lost 'there', not right now, at this river decision we are discussing.

EV matters in every decision, and the EV lost by putting money in the pot during those earlier moments was just negative EV in those decisions, not in this one on the river! We cannot count that double. The EV of folding is zero in this river decision, and always for that matter!

Checking

Now let's wrap our heads around the EV of checking. For simplicity, we are going to assume that when we check, the hand is simply checked down (both players check).

Well, now we can actually win our hand, and the pot, because we get to showdown the winner here. Since we have the nuts with that Royal Flush, we will win the pot every time. We always win $80! That seems great. Our EV is now $80, certainly better than the $0 for folding!

A little heads up here—no pun intended—if we did not have the nuts but had some chance of winning that was less than 100% but more than 0%, our EV would indeed be somewhere between $0 and $80. For instance, if we had a 50% chance of winning, our EV for checking would then also be 50% of the pot, or $40 (0.5 * $80).

We could then say that the relationship between your chance of winning and the expected value is direct and proportional, or 'linear,' in math terms. This means that as your chances of winning increase, the expected value increases at a consistent rate.

Betting

Moving on now to the most interesting decision: clearly, the right decision is to bet with the nuts! Let's be clear here and really drive a point home. If we bet, and our opponent always folds, our EV is the same as checking: $80. But what if our opponent would always call our bet, no matter what we bet? What should our bet size be?

However, how would we know if he would always call our all in here? Perhaps he will call more often if we bet $50 instead, or half our stack…

To really drive home the concept of EV in poker, let's lay out one more scenario:

We see here that our opponent calls more often (80%) if we bet $50 instead of going all in with $100. In this case, which option has the higher EV?

Expected Value (EV) in One Simple Sentence

This all was a very verbose and lengthy explanation—well, call it a rant—that tried to give a mathematical foundation to what EV really is. But if I had to explain it in one sentence, it would be this:

The average single value you expect to get, given multiple potential outcomes.

Taking that simple sentence, along with everything that came before it, this is my best explanation of EV in poker. If this still doesn't make any sense to you, I suggest you read it again. If after three tries you're still clueless about what EV means, perhaps you should consider playing poker just for fun, which is perfectly fine.

In fact, you could consider finishing this course anyway, as in the next articles, we are going to teach practical ways to better your poker game. You can still just play poker for fun, but hey, it can't hurt to win more and lose less! And also, take some time to choose the correct poker room, as it this could also help managing your losses, or optimizing your wins!

How to Estimate Expected Values (EVs) in Poker

Let's not make things prettier than it is, then. Poker is hard. Estimating EVs is hard. Winning at poker is hard. I wouldn't do you, or the game, any justice by making it seem easier than it is. Poker is a study, a full-time study to win a lot. Perhaps a part-time study to win a little.

Winning consistently in poker is about being consistent with your studies. It's like learning a language, a second one, that is. If you don't stay on it, you lose track, you lose your fluency. To become fluent in poker, you need to study hard, maybe for years, in order to win at any meaningful stakes. For the micro-stakes, it can be a lot easier; getting the basics right can be enough.

In the upcoming articles, we will start diving into the basics. These are going to be tools in your arsenal for trying to estimate the EVs of your decisions, or at least in trying to pick the decisions with the highest EV. Remember, you don't necessarily need to get those numbers perfectly estimated; as long as you consistently choose decisions with the highest EV attached to them, you will win just as much as if you were actually getting those EVs pinpointed to the exact right numbers. Let me reiterate:

It is more about choosing the actions with the highest EV than correctly estimating the EVs of all actions.

You don't need to perfectly approach the correct EV values for each action. It may actually be sufficient to just develop an intuition for identifying the action with the highest EV, rather than precisely estimating the EVs for all possible actions every time.

There are tools, heuristics, and many concepts to learn along the way to sharpen your intuition and knowledge, guiding you in making optimal decisions. It's about making the best decisions, not about knowing the exact EVs; remember, we are not Rainman—nobody is. No poker god, not even Phil Ivey, Linus Loeliger, or Daniel Negreanu, gets the EVs for every decision pinpointed to the decimal. They may sometimes come very close, but most of the time, even they can't be completely sure. However, they all focus on making the optimal decisions by choosing actions with the highest Expected Value.

In upcoming articles, we'll delve into the foundational concepts you should begin to understand to grow as a poker player. Establishing a solid base of heuristics, tools, and a thorough understanding of key poker (or math) concepts will go a long way, helping you increasingly choose actions with the highest EV, more and more over time.

Click here to go to the next article, where we will get practical and start talking about the preflop betting round. See you there!