Expected Value

Expected value has an important role in determining the profitability of a player’s poker strategy. Players use this concept to ensure they’re making decisions that are profitable in the long run, and it is a great way to understand the real-money value of your decisions, even if they don’t work out in your favor.

In this article, we’ll be looking at how to calculate EV, how it’s used in cash games and tournaments, and much more to help you master the topic and take your poker game to the next level.

The Foundation of Expected Value

Expected value is a mathematical approach to determining the value of your decisions at the poker table. The expected value formula gives you an insight into the average profitability of your decision if you were to play it out hundreds or thousands of times. A positive expected value represents a profitable decision and a negative expected value represents an unprofitable decision.

Without running expected value calculations, players can only guess at whether their decision would make them money in the long run. By using these calculations regularly, you can assess the profitability of your decision-making, focusing on the decisions that will make you money and eliminating those that don’t. 

Players must consider several factors when determining the EV of their decision, such as pot odds, their equity, their opponent’s range, as well as possible future outcomes. These calculations can get complicated, but we’re here to walk you through the different EV calculations you can make. Once you gain mastery of expected value, you can use it as a guiding principle in your strategic decision-making across multiple poker variants, as the calculations remain the same no matter what poker game you are playing.

Let’s take a look at how to find expected value when making a poker decision.

Calculating Expected Value: What Goes Into The Equation

Calculating your expected value requires you to make accurate assessments of your opponent’s range, their likely actions, and how likely you are to win the hand (for more information on this check out our Poker Odds page). 

It’s considered by many players to be more of an art than a science, as it relies on your ability to make assumptions about your opponent’s behavior to fill in the variables. A slight change in these variables can give vastly different results, so the more accurate these assumptions are, the more accurate your EV calculations are going to be.

To explain what we mean, let’s take a look at the basic EV equation.

Basic EV = ($ Won * Win %) – ($ Lost * Lose %)

What this equation does is calculate how much we lose on average when we lose the hand and takes it away from how much we win on average when we win the hand. This is the simplest EV equation we’ll use, but even this one relies on assumptions about the hand. You need to accurately assess how often you’ll win the hand and how often you’ll lose the hand to get an accurate EV result.

Let’s take a look at an example. We’re on the river with AsKd and a board of AhQh8d6c4s. Our opponent has bet $100 into a $100 pot. We’re considering a call, but we want to determine the expected value of our call. To do this, we need to estimate how often we win the hand when we call, and how often we lose the hand. 

Doing this requires you to take into account your opponent’s preflop range, the betting action on previous streets, their overall player tendencies, and any other factors that may change the way they play the hand.

For the purposes of this example, we assume that we’ll win the hand 75% of the time and lose 25% of the time. Let’s plug these numbers into our equation 

Basic EV = ($200 * 0.75) – ($100 * 0.25)

This shows that we win the $200 pot 75% of the time (the initial $100 pot plus our opponent’s $100 bet) and that we lose our $100 call 25% of the time.

Basic EV = $150 – $25 = $125

The result of our calculation shows that our EV of this call is $125. This shows us what we would expect to win on average if we played this situation hundreds or thousands of times. A positive EV means that we have a profitable action, so we should make the call.

If the result was a negative value, that would mean we would have an unprofitable action, so we shouldn’t make the call.

Pre-Flop Expected Value: Hand Selection and Aggression

While many people associate the concept of EV with postflop decisions, it permeates every aspect of your strategy – including preflop. However, performing EV calculations on preflop hands is incredibly complicated, and is best left to the computers! But that doesn’t mean we can’t use these concepts to shape our preflop decision-making.

There is a direct correlation between the strength of your hand and your position at the table with the expected value of a preflop hand. 

1. The stronger it is, the higher the EV
2. The closer you are to the button, the higher the EV

For example, pocket aces on the button is the preflop hand with the highest EV, as it’s the strongest hand in the strongest position.

It’s important to consider these factors when creating our preflop ranges, as the EV of hands will drastically change depending on the variables of the situation. Certain hands, such as low suited connectors and weak Ax hands have a negative expected value from early position but have a positive expected value from late position. This is why you should always be aware of your how your hand interacts with your position when creating your preflop ranges.

Your preflop position isn’t the only factor that should influence your preflop hand selection. Other variables such as table dynamics, the skill level of your opponents, and your opponent’s play styles should all impact the hands you choose to play preflop. For example, in a situation where all your opponents are playing tight, you should widen your preflop range to take advantage of this and steal the blinds. In these situations, hands that are marginally losing or breaking even become profitable, as you won’t be forced to play preflop with these hands as often.

Post-Flop Expected Value: Navigating Complex Situations

It’s a good rule of thumb that the later into the hand you get, the easier it is to calculate your expected value. When you’re on the river, there are no more cards to come, so the only thing you need to calculate is how much you win when you win, and how much you lose when you lose. However, when you’re calculating your EV on earlier streets, you should factor in future events to your equation to get a more precise result.

Calculating Your EV Across Multiple Streets

The ideal way to do this would be to calculate your EV across multiple streets to get the most precise outcome. However, players have found this to be unreasonable, even for simple situations such as calling the flop with a flush draw.

For example, you’re on the flop with 6h5h on a board of Ah9hKc and your opponent has bet $10 into a $30, and you want to calculate the expected value of a call. To do this, you need to calculate a large number of variables, such as:

• Your chance of hitting your flush on the turn
• The stack sizes in play
• Your opponent’s likely action on the turn
• How often you see a river
• The chance of making your hand on the river
• The value you’d extract if you do make your hand

Calculating all of that in one equation is just too complicated, so players simplify their EV calculations to focus on one street at a time. That, combined with implied odds, gives them an accurate assessment of their profitability from street to street.

Calculating EV Street By Street

Given the above example, we’d use the following equation to calculate the EV of your flop call.

EV = ($40 * 0.18) – ($10 * 0.82)

EV = $7.20 – $8.20 = -$1

After performing this calculation, we can see that the EV of calling in this situation is -$1. However, this equation only considers the profitability of your flop call and does not take into account action on future streets. 

To factor this into our calculations, we must look at implied odds. We have a whole other article on this topic, so we won’t go into too much detail here; the only thing we need to know is that given these variables, you need to make $6 on later streets to break even on this call. If you think you can make this much when you make your hand, you should make the call; otherwise, you should fold.

Let’s say you make the call and the turn comes the 4c – you miss your flush draw. Your opponent now bets $50 into a $50 pot, and again you want to evaluate the expected value of your call.

EV = (100 * 0.18) – ($50 * 0.82)

EV = $18 – $41 = -$23

As we can see, the expected value of this call is much worse than the expected value of our flop call. In this scenario, you’d need to make another $128 on the river to break even on this call, which is possible, but unlikely. Therefore, the sensible decision would be to fold your hand.

This highlights the fact that you should consider the EV of your decisions throughout the hand; just because you made a profitable call on the flop, it doesn’t mean you can see a river no matter what. Being able to accurately asses your EV across multiple streets will help you bail out of hands when they are no longer profitable, which helps to preserve your stack and lets you live to fight another day.

Expected Value and Bluffing: Balancing Risk and Reward

Calculating the EV of your decision doesn’t just apply to calling; you can calculate the EV of bluffing. There are two types of bluff a player can make and each one has a different EV equation: outright bluffing and semi-bluffing. An outright bluff is a bluff that has no chance of winning, whereas a semi-bluff is a bluff where the bettor has a chance of making the best hand by the river.

Outright Bluffing EV Calculations

Let’s start with the easiest one. You’re on the river with 5c4c and a board of Kd9c8cJs2d; you missed your flush draw and you’re considering betting $50 into a $100 pot as a bluff. However, before you do, you want to calculate the EV of your bluff. To do this, you need to perform the following equation.

Bluffing EV = (Pot Size * Fold %) – (Bet Size * Call %)

As part of filling in the variables of this equation, you need to estimate how often our opponent folds and how often they call your bet. This requires you to evaluate the strength of their range based on the action of previous streets, their preflop position, their play style, and other psychological and emotional factors.

For the sake of this example, we think our opponent will fold 25% of the time and call 75% of the time.

Bluffing EV = ($100 * 0.25) – ($50 * 0.75)

Bluffing EV = $25 – $37.50 = -$12.50

As you can see, the EV of this decision is a negative one, which means we do not have a profitable bluff.

Semi-Bluffing EV Calculations

But what if we’re earlier in the hand and have a chance of winning? Say we’re on the turn and have a roughly 18% chance of making the best hand and we want to bet $25 into a $50 pot. How would we calculate the EV of this bluff? We would have to use this equation.

Semi-Bluff EV = ((Pot Size * Fold %) + (Bet Size * (Call % * Win %)) – (Call % * (Bet Size * Lose %))

This equation takes into account the equity we have in the hand and factors that into our expected value. Assuming the same fold and call percentages of our opponent, let’s see how this equation looks.

Semi-Bluff EV = (($50 * 0.25) + ($25 * (0.75 * 0.18)) – (0.75 * ($25 * 0.82))

Semi-Bluff EV = ($12.50 + ($25 * 0.135)) – (0.75 * $20.50)

Semi-Bluff EV = ($12.50 + $3.37) – $15.37

Semi-Bluff EV = $15.37 – $15.37 = $0

As you can see, the EV of our turn bluff is exactly zero, which means the bet is breakeven. This is a significant improvement on the -$12.50 expected value of our river bluff, which shows how important it is to bluff with equity whenever possible!

Common Mistakes in EV Calculations

Making EV calculations can be tricky, and it’s common for players to make simple mistakes when performing the equation. Calculating expected value relies on accurately assessing unknown variables to arrive at an outcome; if you fail to make accurate assessments, your EV calculations won’t be accurate either. 

To give you a leg-up on your opponents, we’ve found the most common mistakes players make when performing EV calculations, so try to avoid these at the tables!