Expected Value

Expected Value is the amount of money an action expects to win or lose on average.
Expected Value

Expected Value (EV) is a term you will come across again and again in forums and in poker strategy articles. In this guide I will explain exactly what expected value is and why it is important when it comes to making decisions at the table.

Although somewhat similar, try not to get expected value confused with equity. If you are getting confused between the two, read the article on the difference between equity and EV.

What is expected value?

Expected value is the amount of money a certain play expects to win or lose on average.

In any given situation in poker, a check, call, bet, raise and fold will have certain expected values. Some of these plays will win money, whereas others will lose you money. Out of the actions that will win you money, some will win more money than others on average.

As you can guess, the aim is to make the play with the greatest expected value.

Here are two abbreviations that you will want to familiarise yourself with:

  • +EV - This is a positive expectation play that will win money over the long run.
  • -EV - This is a negative expectation play that will lose money over the long run.

How to work out expected value.

You multiply the results of the possible outcomes by their probability of happening, and then you add all of them together.

Trust me; it’s really not as difficult as I make it sound. Mathematics always looks a lot harder before it’s explained to you, so let me give you some examples.

  1. Simple coin flip example ↓
  2. Simple poker example (using a flush draw) ↓

1) Simple coin flip example.

Almost every good article or guide that attempts to explain expected value uses a straightforward coin-flipping example to start off with, so I’m not going to be breaking any trends here. Let’s get started...

  • A friend offers to pay you $1.50 every time she flips a coin and it lands on tails.
  • However, every time it lands on heads you have to pay her $1.

What is the expected value of each and every coin flip? How much do we expect to win or lose on each individual flip? Is it a profitable game for us?

To work out our expected value for this game, we need to look at the results of each possible outcome and their probability of happening.

Possible results and their probabilities:

Heads = -$1.00
    p(Heads) = 0.5

Tails = +$1.50
    p(Tails) = 0.5

p(Heads) is a faster way of writing "the probability of heads". Guess what p(Tails) means.

If it’s a fair coin (as opposed to all those unfair coins you run in to), the probability of it landing either heads or tails is 1/2, or 0.5.

All we have to do now is multiply these outcomes (the amount we win in each possible outcome) by their probability and add them together to find the EV for each coin flip.

Working out EV for each coin flip.
                        
EV =  Heads      +  Tails
   = (-$1 x 0.5) + ($1.50 x 0.5)
   = (-0.5) + (0.75)
   = $0.25

This means that every time we flip a coin in this game we are winning $0.25 on average. Over 2 flips we should win $1.50 once and lose $1 once, given us a net profit of $0.5 over 2 flips. Therefore, over 1 flip this works out to earn us $0.25 on average.

It doesn’t make a difference if we lose 10 flips in a row, because over the long run this will remain as a profitable game (unless our good lady friend has decided to cheat us in some way of course).

There will be variance, but over a big enough number of trials the amount we have won should be very close to our expected value for those trials.

2) A basic example with a flush draw.

It’s all well and good learning about the basics of expected value with coin flips, but how does expected value apply to poker? It’s pretty much the same thing, so let’s look at a straightforward expected value example with a flush draw.

Our hand: As 2s
Board: Qs Kh 3s 7d

The pot is $100 and our opponent moves all-in for $50. So we have to call $50 for a chance of winning a total of $150. Assuming that the only way for us to win the hand is by hitting our flush on the last card, what is the expected value of calling? In other words, is it profitable for us to call?

We can work out if calling is profitable using pot odds, but with expected value we’re going to work out exactly how much we expect to win or lose on average by making the call.

Possible results and their probabilities:

Call, hit flush   = +$150
    p(hit flush)  = 0.2

Call, miss flush  = -$50
    p(miss flush) = 0.8

The probability of hitting a flush on the river is 4.1 to 1, which is roughly 20% chance or 0.2. Therefore, the odds of not hitting a flush will be 1 - 0.2 = 0.8. I used the outs odds charts for these particular odds.

Also, notice how we are looking to win $150 and only lose $50 in each outcome. We are only going to lose $50 because that is how much we are paying to try and hit our flush in this single decision. We are not factoring in money that we have put in to the pot in previous betting rounds. We just take the facts from this decision alone.

Working out EV for calling:

EV =  hit flush   +  miss flush
   = ($150 x 0.2) + (-$50 x 0.8)
   = ($30)       + (-$40)
   = -$10

This means that every time we call this bet to try and hit our flush, we are losing $10 on average. Therefore this is a -EV play and we should fold instead of call.

What is the use of expected value in poker?

Every single play you make in poker revolves around the concept of maximizing your expected value.

If you can always manage to make the play with the greatest expected value, then you will be able to win the most money possible from each and every session you play.

Obviously making the most +EV plays in every single situation is not going to be possible for most, but it is something that you should strive to achieve even if you never expect to reach this goal (as bleak as that sounds).

Good poker strategy revolves around making +EV decisions.

There is a wealth of topics and guides when it comes to poker strategy (see the Texas Hold’em strategy section on this site for example). At the core of each of these tips and strategies is the aim to help you make the most +EV plays possible and help you to avoid making -EV plays. That’s basically what all poker strategy is about.

How do I use expected value during play?

You don’t to be honest. Expected value is not like pot odds in that you can use it on the fly to work out whether decisions are profitable or not. You simply do not have enough time to work out the EV of every possible play to help you find the most profitable action.

Expected value is best used for post-game analysis where you try and work out whether or not you had made optimal plays in certain hands. EV is also a very important concept that helps to explain why some plays are good and why some plays are bad.

Evaluation.

Expected value in poker is the amount of money you expect to win or lose from each play you make. The more +EV plays you make, the more money you will win. It’s as simple as that.

Expected value (or EV as you will come to familiarise yourself with) isn’t really a topic that is going to revolutionize your game right now, but nonetheless it’s definitely one of the most important mathematical concepts to learn about. Working out EV in hands can get a lot more complicated than the ones above, but the process for working out expected value is exactly the same.

If you’re familiar with the REM process, the maximize section is all about maximizing value from your hands, which is a key component to making money from poker.

If this article didn’t quite do it for you when it comes to working out EV, try my "boxes method" for calculating expected value in poker. You may find that more helpful.

Go back to the awesome Texas Hold'em Strategy.

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