Ev Formula Poker

The idea behind poker, especially tournament poker, is to maximize your win when the opportunity presents itself. Simple EV can 'leave money on the table' so to speak. If you can maximize your EV you'll, over time, simply win more money/chip is a shorter period of time. And money/chips over time is how we keep score in poker. The basic formula that is used to determine enterprise value is as follows: EV = CS + PS + MVD + MI – CE. Where: EV is enterprise value, CS is the market value of common shares (market capitalization), PS is the market value of preferred shares, MVD is the market value of debt, MI is minority interest, and CE is cash and cash equivalents.

As a poker player you will have experienced times when you made the correct decision only to have the results make you want to punch the wall. Poker is a game with variance, meaning that things are going to happen that go against the odds of them happening (sometimes seeming like they defy all possibility). However, as long as you are making decisions that have a positive expected value (+EV), you will be profitable in the long run.

What exactly does expected value mean? Basically, if you were to flip a coin and someone offered you $1 for every time you called heads or tails correctly and there were no penalty for guessing incorrectly, that would have a positive expected value. Since there is an even 50/50 chance of earning $1 each flip, you can say that the EV of each flip of the coin is + $0.50.

Now if you wagered $0.50 each flip, it would be an even money bet. Half of the time, you would lose $0.50 and you would profit $0.50 the other half ($1 of winnings minus the original $0.50 wager).

Expected value—commonly referred to as EV—is the long-term result of your decisions in a particular poker hand. It is your way to cut through poker’s blend of luck and strategy so you are able to see how profitable your decisions are. A simple example of Expected Value (EV) put into practice - if you were to bet $10 on heads in a coin toss, and you were to receive $11 every time you got it right, the EV would be 0.5. This means that if you were to make the same bet on heads over and over again, you can expect to win an average of $0.50 for each bet of $10.

Now let’s say you are given $1 if you call a coin flip correctly, but you have to wager $0.55 each time you flip the coin. You would not want to make this bet because it has a negative expected value (-EV). Over time, you would lose an average of $0.05 on each flip.

Playing Poker for the “Long Term”

Using the same coin flip analogy, in a sample size of 10 trials there are going to be times when heads comes up all 10 times. You know each time you flip the coin the odds of it coming up heads are 50%, but over the course of this relatively small number of flips the results seem to defy the odds. However, if you extend the number of flips to 100 or 1000 you will get closer to an accurate result of 50/50.

So that is what the “long term” is in poker. If you make the same play over an infinite number of trials, the resulting amount of chips you earn over time is either going to be positive or negative. Because the distribution of cards is random, there is rarely a guarantee that you will have the winning hand until all the cards are dealt. So in any given hand there is always a probability that the best hand can end up losing and the actual results deviate from the way the odds say they should. In hold’em, a pair of Aces is about an 80% favourite versus a pair of Kings pre-flop. That still means that 20% of the time the pair of Kings will win. But in the long run the Aces have a positive expectation to win.

Getting the Most Value Out of Hands

Expected value is not just about a singular outcome (winning vs. losing), but it’s also about maximizing the value in every hand you play. This means squeezing the most value out of hands when you are ahead and losing the least when you are behind. It can also mean knowing how much to risk on a bluff where the percentage of times it will work and the value of the pot makes it profitable in the long run.

How to Squeeze Maximum Value

Let’s look at a hand example to illustrate…

• No Limit Hold’em Tournament Play
• Blinds: 25/50
• You and your main opponent both have 3,000 in chips.

Pre-Flop

The following hand example will demonstrate how to squeeze maximum value against a player on a suspected drawing hand. Importantly, you already know that this player likes to play suited cards and likes to chase draws. Before the flop, your opponent in middle position limps and you elect to call from the button with . The small blind completes and the big blind checks.

The Flop

The flop is dealt and everyone checks to you, as shown in figure 1 below:

Figure 1

Jackpot, a set! But wait – what if someone has a flush draw?! Should you be afraid of someone drawing to another heart and try and shut the hand down right now by putting in a big bet? No, you shouldn’t. This is a very common mistake. Many players see the potential flush draw and overbet the pot or push all-in to “protect their hand”. Protection is incorrect thinking because it doesn’t maximize your expected value. If you try to shut the hand down now with a big bet you will lose money in the long run. Does this mean you should slow play by only betting a small amount or even checking? No, it doesn’t. It means that you should bet the maximum amount that you think someone will call to draw to their hand and make an incorrect decision.

For someone to correctly draw to the flush with one card to come, they need 4-to-1 odds. So in this case, the pot is 200. If you bet 50 trying to slow play, the pot will now have 250 and it will cost them 50 to call giving them 5-to-1 odds (250/50 = 5/1). This would be a correct decision for them to call and would be –EV for you because they will hit their draw better than the 5-to-1 odds you are giving them. So, what is the correct amount to bet? This depends on your opponent. If you know someone likes to chase draws, you should consider betting around the size of the pot. If you know they will only call a reasonable sized bet, then you should bet enough to give them about 3-to-1 odds. This would be an incorrect call on their part.

There are additional factors of implied odds to consider here, but that is for another lesson. In this case, you know at least one of your three opponents likes to chase draws, but you can’t be sure he has one now. I would bet around 100. This will make the size of the pot 300 and will give someone 3-to-1 odds to call. The guy in middle position who likes to chase, calls.

The Turn

The turn is a ten of spades and our opponent checks:

Figure 2

How much should you bet here on the turn to maximize EV? Well, you can be pretty confident that your opponent doesn’t have a King because they would have likely bet the flop against three opponents. While they could have a straight draw if they had a hand like 45 or 34, the most likely hand your opponent has contains two hearts including AX and connectors such as or . If he does have , he now has a pretty big draw and is unlikely to consider folding. But we don’t know that for sure, so we have to bet an amount that maximizes our EV against a range of drawing hands. The pot is now 400 and our opponent has 2,850 left. Based on our opponents tendencies, I would bet around the size of the pot, so let’s make it 350. Our opponent calls.

The River

The river is a beautiful . If our opponent was on the flush draw, he just hit, but it also gives us a full house. The size of the pot is 1,100 and our opponent bets out 600:

Figure 3

His bet lets us know that he likely has the flush. He could also have a King, but it’s not likely based on his previous actions. Either way, we’re in a great situation.

Now, how do we get the most out of this hand? After his bet of 600, there would be 1,700 in the pot and it leaves him with 1,900. Many beginning players will raise the minimum here because they are afraid of making their opponent fold. But that is leaving money on the table.

Players who often chase draws will not fold when they make their hand. They feel emotionally attached because they have already spent a lot of their stack to get there. Also, we have 1,900 left and if we just raise the minimum to 1,200 we are committing most of our stack which looks like we have a huge hand. I would think for a few seconds and then push all-in. Our hand is pretty concealed and it looks like we have a King, so it’s highly likely we’ll get called here. Our opponent calls showing and we rake in a monster pot.

Don’t be Results Focused

Since you cannot control the final outcome of any given hand, the goal in poker is not to win every hand, but to make decisions that have a profitable expected value. Sometimes luck is in your favour and sometimes it’s against you, but if you are making +EV decisions that is what makes you money in the long run. It is important not to let negative results get you down or hurt your confidence in your abilities. Just remind yourself that you wanted that donkey to call you down with bottom pair because even though he spiked two-pair on the river this time, he is your personal ATM if he keeps making that play.

A firm grasp of the concept of expected value will serve you well. In our next lesson, calculating EV, we’ll take things a step further and discuss the additional criteria that must be incorporated into your decisions. We’ll also look at some common EV spots in both cash games and tournament poker – all with the intention of positively affecting the long-term profitability of your decisions.

By Donovan Panone

Donovan started playing poker in 2004 and is an experienced tournament and cash game player who has a passion for teaching and helping others improve their game.

Why are people losing at the casinos? Why shouldn’t you buy a lottery ticket? How do you account for uncertainty when you invest a smaller or bigger amount of money? And what should you consider when you calculate the ROI of a data science project? Behind all these questions there is one powerful statistical concept: expected value!

In this article, I’ll show you:

What is Expected Value?

Expected value is a theoretical value that shows the average return of an action you’d get if it was repeated infinite times. You can calculate expected value as the weighted average of all the possible outcome values — where the weight is the probability of the given outcome.

I know, I know… on the first read, this sounds complicated. But believe me, it’s not. Let me give you a simple example and everything will fall into place immediately.

A friendly game

You and your friend play a game. Your friend has a hat with 10 balls in it:

• 5 blue balls
• 4 yellow balls
• 1 red ball

You draw one ball from the hat. If you draw:

• a blue ball, you’ll win $0
• a yellow ball, you’ll win $2
• a red ball, you’ll win $10

Let’s calculate the expected value of this game!

Take all the possible outcomes and calculate their weighted average — where the weight is the probability of the given outcome. For your convenience, I put all the details into one table:

The calculation goes:

So the expected value of this game is: $1.80.

In other words if you played it long enough, let’s say for 10,000 rounds, you’d end up with something pretty close to $18,000 (which is 10,000 * $1.80, you know).

Obviously, if you played only one round, you’d get $10, $2 or $0… and not $1.80.
As I said: expected value is a theoretical value. But it shows itself on bigger sample sizes in practice, too.

A less friendly game

In the previous example you played with a friend. She didn’t ask you to risk your money. You could only win. How nice of her! In real life though, it’s more likely that you’ll have to pay a fee to get into the game.

Good news!
Now that you know the expected value of this game ($1.80) you can immediately tell how much money you can risk to stay profitable in the long term.

Your expected value calculation changes like this:

The only new variable is the entrance fee, of course.

• If it’s exactly $1.80, your expected revenue in this game is $0,
• if it’s less than $1.80, you make profit,
• if it’s more than $1.80, you lose money

…in the long term.

As I said, the concept of expected value is so, so simple. And that’s why my mind is always blown when I see people ignore it in so many parts of their life.

The Expected Value Formula

The expected value formula is this:

E(x) = x₁ *P(x₁) + x₂ *P(x₂) + x₃ *P(x₃)…

• x is the outcome of the event
• P(x) is the probability of the event occurring

You can have as many xz * P(xz)s in the equation as there are possible outcomes for the action you’re examining.

There is a short form for the expected value formula, too.

Ev Formula Poker Rules

E(x) = ∑x * P(x)

The formula, by the way, shows the same thing you have seen in the examples before: it’s the weighted mean of the possible outcomes, where the weight is the probability of each event occurring.

Examples of applying and calculating Expected Value

Let me give you a few more real-life examples to hammer home the concept and the math!

Example #1 – Coin flip

What is the most fair gamble in the world? Flipping a coin!
You have two outcomes: heads or tails. The probabilities of both are 50%.

Let’s say that you play 100 rounds with your friend. You risk $1 in each round. If it’s tails, you double your money, if it’s heads, you lose your money.

Using the expected value formula:

The expected revenue from this game is $1. And you have to invest$1 in each round. So your expected value of your profit is $0. In other words, if you play this game long enough, you won’t lose or win any money.

Okay, so this is the theory. But does it work out in practice?
Let’s run a simulation to discover that!

Here’s a visual!

• The orange line represents the expected value in each round. This is the theoretical value. Luck is eliminated. Again, it’s always $0 because your investment ($1) equals your expected revenue ($1).
• The blue line is the real stack. It has a natural variance. It goes up and down, depending whether you were lucky (you got heads) or unlucky (you got tails).

As you can see, the expected value was $0 – but you ended up with $5 after all. Yeah, this happens, you know, it’s called blind luck. But calculating the expected value helps rationalize that.

Speaking of luck…

Most people misinterpret the probability of improbable things. Here’s the same game, the same simulation, the same fair coin — but over 10,000 rounds this time. And look at that lucky run between round #3000 and #5000. This is natural variance in action, again. The word natural fits well in this situation because seeing a fluctuation like this in real life is totally normal.

Interestingly enough, it goes back to 0, after all.
That’s called the central tendency and the more you play, the more it applies.

Example #2 – Roulette (black vs. red)

I never play roulette.
Why? Because I know that the more I play, the higher the chance that I’ll lose. But how much exactly?

Just apply the expected value formula here, too.

Let’s say that you want to put $1 on black.

What’s your expected value?
The outcomes:

• black: probability 18/37, you win $2
• red: probability 18/37, you get nothing
• zero: probability 1/37, you get nothing

Using the expected value formula:

So $0.97 is the expected revenue. Given that you invest $1, your expected profit is -$0.03… so in theory, you lose 3 cents in each round.

Let’s see the 10,000-round simulation of this one! It’s really sobering:

Ev Formula Poker Strategy

• the orange line shows the expected value of your stack (theoretical value)
• the blue line shows the real value of your stack (luck and natural variance involved)

In this particular simulation, we were very lucky because we ended up above the expected value. Yet with a $200 loss.

Oh, and if you think I went with the example that serves my message, here’s the next six simulations I ran right after this one:

Expected value and central tendency is powerful.
As they say: the house always wins. (Check out my new Youtube video on the topic: Why You Shouldn’t Go to Casinos… you can do it in podcast format, as well.)

Example #3 – “Risk-free” investments

There is no such a thing as risk-free investment.

There are low-risk investments and high-risk investments. And if you are smart enough, you can pick a low-risk investment with a high enough expected value.

But again, all investments involve some risk. And you should account for that before you put your money (or any other resources) into it.

Here’s a simple example:
Most European countries offer government bonds. They usually pay ~4% interest per year. And they are considered to be extremely secure investments. After all, countries don’t go bankrupt very often, right? (Sometimes they do though.)

But let’s run the numbers!

You want to invest €100,000 and you’d realize a 4% yield after one year.
If there were no risk at all, your expected value would be simply:

So it’s a €4,000 profit. Yay.

Ev Formula Poker

But you have to account for the potential risks, too!
Let’s say there’s a marginal chance that the country goes bankrupt and you lose all your money (again: it’s improbable but can happen). Set an extremely low probability for that: 0.01%.

Your expected value formula changes this way:

Okay, it seems that we still have a very good expected value. Country bankruptcy is not a significant factor. Great!

Another risk is that you might need your money and take it out earlier than in 1 year. In that case, you’d lose the yield and usually, you’d have to pay a penalty, too. The usual penalty rate is ~2%. The unknown variable is the probability that you’ll have to take out your money — let’s go with an estimated value: 20%.

So the EV calculation goes:

Still a positive value — although €2,789.6 is much lower than the original €4,000.

Note: And we haven’t even considered inflation, opportunity cost, and so on…

Are government bonds good or bad investments? I don’t care — this is not a money blog. 🙂 All I’m saying is that before any investment, you have to run your numbers, account for all possible outcomes — and calculate expected value to have a realistic picture.

The hard part: guessing probability (stock market, poker, etc.)

Applying the expected value formula is simple. Knowing all the variables in it is the hard part.
Especially the probability of the specific events. Even in that simpler bond-investment example above, I had to go with estimates and guesses — because I don’t have solid information on the likelihood of a country going bankrupt.

But that’s fine. Sometimes you have clear numbers and it’s easier to make the right call (e.g. not playing roulette). In other cases, you don’t. Regardless, in these cases, your goal is to collect as much information as you can and come up with estimates that are as realistic as possible.

Note: A good example can be playing poker. You know what’s in your hand. And that’s important information – you can already calculate your chances based on that. But you can improve your math if you can narrow down what could be in your opponents’ hands. Real poker pros know all these tricks and it’s not an accident that they win more than others.

Ev Formula Poker Calculator

How to use expected value in your everyday life

If you think expected value is a new concept or that you can use it in data science only, let me mention that the great Blaise Pascal tried to use it to argue whether it’s worth it to believe in God or not. 🙂 Well, that’s an extreme (and maybe not the best) application of the formula. But it shows very well that statistics also has its philosophical depths. For me, starting to apply expected value in my life was a true mindshift. I realized that nothing is certain, but most things have a high enough probability and reward to take a risk.

E.g. quitting your full-time job and starting your own company instead. Is it a good or a bad financial decision? You just have to estimate your outcomes and their probabilities.

Using the expected value formula:

Again, I just came up with these numbers, they differ from person to person. And I know this is an oversimplification, too. But the point is: using expected value as a concept in your everyday life can help you to rationalize emotionally stressful and/or scary decisions.

It can also help you to avoid bad decisions.

Note: Homework! Is it worth speeding on highways? Try to run the expected value calculation by yourself! (Hint: How much time do you save by driving at 150 kmph instead of 120 kmph? 10 minutes? 20 minutes? And what’s the probability that you’ll die and lose 20 years or 30 years on the other hand? What’s the expected value of speeding?)

Okay, so before we go too deep into these philosophical questions, let me answer a more data science related one, too…

How to use expected value in your data science career

When it comes to data science, you can take advantage of expected value in (at least) two ways.

First, you can use it directly in any situation where you are working with probability values. E.g:

• You classify users as potential buyers with 80% probability. Is it worth spending money on reaching out to them? The expected value formula can help you with the answer…
• You run an e-commerce store selling fast-moving consumer goods (FMCG). What’s better: running out of your top-selling product from time to time vs. overstocking it and accepting that you’ll have waste from time to time?
• Your new version in an A/B test reached only a 90% statistical significance. Is it worth the risk to go with it, regardless? (Hint: usually it isn’t.)
• …

And secondly, you can try to calculate whether it’s worth running a given data science project at all. What will be the return on the time you invest on that project? What’s the probability that you’ll get the results that you are aiming for? These are, of course, again questions where answers need a lot of guesswork. But even with a ballpark estimate, you can rationalize your decisions and say yes or no to a project idea with more certainty.

Test yourself! How rational of an investor are you?

Applying the concept of expected value in a simpler money decision should be easy. But I learned that it isn’t for everyone. It takes time and experience to get good at it. So I created a little online game to help you practice. Check it out and figure out how good of an investor you are. (At the end of the game you’ll see where you are ranking compared to all other players.)

Check it out here: https://bestbet.data36.com/

Conclusion

I know, folks, not everything has to be rationalized, formulatized and calculated. But the concept of expected value will come in handy so many times in your life and in your career! Especially when you’ll have to make big decisions. So use it to:

1. List all the different outcomes of a decision,
2. Get (or try to estimate the) probability each of these outcomes,
3. Run the expected value formula as you learned it,
4. And make a better decision!

Ev Formula Poker

• If you want to learn more about how to become a data scientist, take my 50-minute video course: How to Become a Data Scientist. (It’s free!)
• Also check out my 6-week online course: The Junior Data Scientist’s First Month video course.

Poker Ev Formula With Fold Equity

Cheers,
Tomi Mester