Poker Expected Value (EV) Formula

Poker EV, short for expected value, is the most vital mathematical concept in poker.  When we say that something is +EV it means the play is expected to be profitable in the long run.  Whereas a play that is -EV is expected to lose us money in the long run.  

The Poker EV Formula

The most simple poker expected value (EV) formula is this:

Not sure what these variables mean?  Either push play to watch this short poker video or read the guide below, which will teach you everything you need to know.

EV is the most important mathematical concept in poker. Without a solid grasp of EV and the ability to create +EV plays, a player is doomed to fail. In this guide, I’ll show you what EV is, how to use it, and why understanding it is vital for poker success.

What Does +EV Mean?

Like I mentioned earlier, EV stands for expected value. It’s the mathematical way of saying “in the long run this play is expected to net me X amount of money”. If you’ve heard the terms +EV or -EV before, these simply describe if a line or play rates to make or lose money in the long term.

Our goal in poker is to consistently make +EV plays. Because EV is mathematical, there is a formula, but it’s not that scary. This is one of the simpler EV equations we will use in poker:

Let’s break it down simply. We have %W which is how often we will win a given hand. We have $W which is how much we will win the times we do in fact win. We have %L, which is how often we will lose this hand. And lastly, we have $L, which is how much money we lose when we lose this hand. Not too bad right? But how can we actually use this? I’m so glad you asked!

Simple EV Examples

You and I are going to play a game. It’s a fun game where we take out a fair coin, with one side heads and one side tails, and we flip it. In this game, if the coin lands on heads I will pay you $3 and if the coin lands on tails you will pay me $1.

If we pull out our EV formula again we can start filling in the variables and solve it. We know that when you win you get $3, so $W = $3, and we know that when I win you lose $1, so $L = $1. We also know that because this is a fair coin there is a 50% chance of it coming up heads and a 50% chance that it comes up tails. So both W% and L% are 50%. Just a quick trick that you can remember is that %W + %L always equals 100%, so if you know one, you always know the other.

So if we do the math quickly we see $1.5 – $.5 = +$1. This means that in the long run you are expected to win $1 each time we flip the coin. Now if we only flip the coin two times your only outcomes are +$6, +$2, or -$2…so you can see how in the short term the results can seem quite different than the +$1 expected value we calculated a second ago. But if we flip the coin millions of times you will average a $1 profit each time I flip.

In poker, we focus on the long run, not the short term. We recognize that the results can vary wildly in small samples, but we know that in the long run the math will bring everything back to its expected value. This means two very important things:

1. We want to constantly find little games like this that are +EV
2. We want to avoid playing games where the EV is negative

With that said, let’s see how this all applies to poker.

A Preflop Poker EV Example

In this hand, it folds around to the small blind who goes all-in. We hold AQ and are debating what we want to do. Now that we are armed with the knowledge of EV we can actually prove this situation mathematically. We just pull out our fancy formula and start plugging in numbers.

So in this situation, we can easily figure out $W and $L. If we call and win we will win the SB’s stack and also our $1 big blind. Once money has been put into the pot, even just a forced bet like the big blind, it no longer belongs to us. That means the $W in this spot is $1 + $12 for $13 total.

The $L is simply how much we would lose if we called this and lost the pot. Well since the $1 big blind doesn’t belong to us we can only lose $11 by calling this. So $L equals $11.

The last thing we need is the %W and %L. In the coin flip example, we knew that a coin had a 50/50 chance of coming up heads or tails. But what about in a poker hand? To figure out our %W and %L we can use an equity calculator like Equilab and figure out our equity (or estimated chance of winning) against our opponent’s range of hands.

For simplicity’s sake, let’s just assume that the SB would shove 77+/AJ+/KQ here. I would normally assign a much wider range, but to make life simpler, let’s just use that range for the time being. We can plug that into an equity calculator like Equilab and we see that our AQ has 47% equity. So we expect when we call here with AQ we will win 47% of the time and lose the other 53%.

Now we just plug everything in, solve it, and ensure that a call here is +EV.

If we look at our two options here, between calling and folding, a fold would be 0EV because we don’t make or lose anything…and a call would be +EV to the tune of $0.28. This means that a call is not only +EV, it is also optimal here.

A Postflop EV Example

You’re playing in a $1/$2 No-limit Texas Hold’em cash game. The blinds are $1/$2, and you and your opponent have effective stacks of $200. The hand reaches the river, and the board reads 2♠️ 5♣️ 8♦️ J♠️ Q♦️. You hold 9♠ 7♠️, which means you missed your straight draw, and there’s no flush on the board. Your opponent is a nit and tends to fold to aggression.

Pot Size: $150 (before any river action)

Your Play: You decide to make a river bet of $100, hoping that your opponent will fold, considering the coordinated board, missed draws, and overcards to any flopped pairs.

Probability of Your Opponent Folding: Based on your observation of your opponent’s playing style, you estimate there’s a 70% chance your opponent will fold to your river bet.

Now, let’s calculate the Expected Value using the formula:

The calculated EV of $75 indicates that, on average, making this river bluff is expected to profit you $75 in the long run. Given your estimated probabilities and the pot size, this positive expected value suggests that the bluff is profitable.

It’s crucial to note that the success of this play depends on your opponent’s tendencies and your ability to accurately assess the likelihood of both their range and folding frequency in this specific situation.

+EV vs. Optimal

An optimal play in poker is when it is the absolute most profitable play you can make…and calling here is optimal against the range we assigned. While $0.28 may not seem like a lot of money, if we consistently call here we will net a significant amount of profit in the long run.

In real-time, you won’t be able to pull out an equity calculator, plug everything into the EV formula, and then solve. But constant practice with solving EV, estimating your equities against various ranges, and simple shortcuts will help you closely eyeball the EV of a play in real-time. And when you have lots of available time when studying, calculate the EV of your plays so that it becomes second nature.

Still Not "Getting" Poker Math?

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An EV Poker Hand Example

Want another example?  Let’s look at a very simple river example where we can plug in variables and solve for the EV.  Take this spot where we decide to bluff the river with a missed draw:

We have enough information to start filling in parts the EV formula.  We know if we ever get action we will lose, so $L is $125.  We know that if the button folds to our bet then we win the pot, so $W is $187.  So the formula is now:

%W in this spot is how often the button would fold against our bet.  An important trick to remember is that %W + %L = 100%.  So if you only know %W you can always figure out %L (and vice versa).  In spots like this you make an assumption of how often the button would fold when you bluff.  You can go really in-depth and do a full combination and frequency analysis of his hand range, but let’s just simplify it here and assume he’d fold 45% of the time.  Now we can solve everything:

So this bet is +EV given that assumption, and of course, if he folds even more than 45% of the time this play just gets more and more profitable!  These kinds of situations are very common in poker and by using EV we can prove the validity of our plays.  

Comparing +EV Lines

The more you study hands, the more often you’ll find situations where there are multiple +EV options available. For instance, you could raise to normal size with AA preflop and that’s +EV, or you could shove all-in preflop with AA, and that would also be +EV. With nutted hands pretty much any aggressive play is +EV, which means it’s important to compare +EV lines correctly.

Let’s go through an example of this by analyzing this overbet I made against a reg in a $1/$2 live cash game. I find myself hitting a great turn & river runout, and opt to compare a couple of different sizing options on the river.

Notice that the variables you assign (primarily your equity vs. their river check/call range and their fold % facing a river bet) are the key drivers in just how +EV your bet is. For extra credit, you could even through a river all-in for $418 and see if that can become the optimal bet size if villain would only call with trips Aces.

Importance Of Making +EV Plays

Making +EV (positive expected value) play regularly in poker is essential to long-term success. In essence, these choices maximize potential gains while minimizing losses. Embracing +EV strategies involves deeply understanding probabilities, opponent tendencies, and optimal decision-making.

The benefits of consistently making +EV plays are profound. Over the long haul, players prioritizing such decisions build a sustainable advantage, steadily growing their bankroll and profits. This disciplined approach mitigates short-term variance, ensuring that the inherent luck factor in poker doesn’t overshadow skill.

On the other hand, long-term profitability is negatively impacted by continually making -EV (negative expected value) plays. -EV plays are choices that, over time, are statistically likely to result in losses. Making such decisions, whether due to impatience, emotional reactions, or a lack of strategic understanding, erodes a player’s bankroll and increases the amount of poker money they would require to play correctly.

This short-sighted approach is akin to making high-risk, low-reward investments jeopardizing long-term financial stability.

Free Poker EV Spreadsheets

If you are interested in calculating the EV automatically, grab my poker EV spreadsheets today. These sheets are incredibly useful for doing these calculations quickly rather than by hand!

My spreadsheets pack includes a basic EV calculator, a complex EV calculator, and the ability to compare EVs quickly (such as comparing the value of betting vs. checking the river).

Practice With EV & Other Math

Want to practice putting your new knowledge and other poker math skills to work? Take my free 20-question quiz and do your best to score 80%+.

Don’t worry if you find it challenging. If you can’t figure out an answer within 30 seconds, just take your best guess and move on as there is a complete answer key available at the end.

Good luck!