What is Expected Value in Poker?

What is Expected Value in Poker?

Expected value in poker is the average return that you estimate an action will bring. Formally, the expected value corresponds to the weighted average of all possible outcomes according to the probability of every outcome occurring.

Calculating the expected value of various possible actions can guide you towards the best one!

A simple example, the Coin-Flipping

Let’s consider the simplest of examples, the coin-flipping. If someone proposed you to play a game of “heads or tails,” where you would win 1$ when the coin lands on heads, and lose 2$ when the coin lands on tails, would you accept? Your intuition says that no! Let’s calculate the expected value of taking that bet to see why.

EV = P(heads) * $(win)- P(tails) * $(lose) = 0.5*1$-0,5*2$ = -0.5$

So, on average, by accepting that bet,  you lose half a dollar every time that you toss the coin. This doesn’t mean that you cannot get lucky and win, but it is an unfavorable bet, and if you repeat it many times, chances are that you will end up loosing!

How can we use Expected Value in poker?

In poker, we have to calculate the weighted outcomes of each possible action, to estimate their expected value. Then, we must compare the expected values of all possible actions (fold, call, raise?) to choose the best one!

To become a winning poker player, you have to choose the actions that have the best +EV. This does not guarantee that each particular hand will turn out favorably. However, in the long run, the chance factor will more or less even out. The player that makes the best +EV decisions ends up with the most money!

a simple example

We will demonstrate this concept with a relatively straightforward situation where your opponent is all-in, and your only options are to call or fold. So, there will be no further action later on the hand.

You are playing heads up with a tight opponent that rarely bets without a solid hand. You have K♥3♥ and on the turn, the board is A♥10♥8♣4♠, so you have the nut flush draw.

There are 100$ in the pot, and your opponent bets his last 50$. You give your opponent credit for at least top pair, so you are only drawing for the flush. What should you do?

Calculating the Expected Value for each option

Your first option is to fold. Your expected value is equal to zero !! Whatever amount you have contributed to the pot is no longer yours. Since you will invest no more money in the pot and expect to gain nothing, your expected value is ZERO.

Your second option is to call. Let’s first calculate the probability that you will win the hand. You need to hit your flush draw to win the pot. This means that out of the 46 cards that are unknown to you, you have 9 outs. The 9 remaining hearts (13 total minus 4 exposed). You have 9 out of 46 or approximately a 20% chance to win. When you win, you gain 150$, the 100$ that is already in the pot plus the 50$ your opponent bet. The remaining 80% of the time that you do not hit your draw, you lose 50$, the amount that you invest by calling your opponent’s bet.

calculating the expected value

Your expected value can be calculated as:
EV (Call)= P(win) * $(win)- P(lose) * $(lose) = 0.2*150$-0,8*50$ = -10$

So on average, if you make the call you will lose 10$. Therefore, the correct choice is to fold since a zero expected value is better than a negative one!

a more complicated scenario

Let’s consider a slightly more complicated case in the above example.

You have K♥3♥ playing against the same tight opponent and on the turn, the board is as before A♥10♥8♣4♠, so you have the nut flush draw. There are 100$ in the pot, and your opponent bets 50$, but this time, your opponent has 150$ more left and you have him covered.

The situation becomes slightly more complicated as you have more choices to consider, and potentially an additional betting round to take into account. You can either fold, call with another betting round to follow after the river, or raise. As we saw before, folding has zero EV. Let’s examine the EV for the other two possibilities.

When you call, you will win about 20% of the time and lose the remaining 80%. However, when you make your draw, you will be able to make one more value bet and potentially extract more value from your hand. Let’s say that you plan to bet 150$ more when you make your flush and estimate that your opponent will call at least 40% of the time. So, the expected value is:

EV (Call)= P(win) * $(win)- P(lose) * $(lose) = 0.2*(150$+0.4*150$)-0,8*50$ = 2$

So, now calling becomes a +EV move.

When raising, you plan to raise to 200$ to put your opponent all-in. You estimate that your opponent will call at about 50% of the time and fold the remaining 50%. So, your expected value is:

EV (Raise)= P(opp.fold)*pot+P(opp.call)*{P(win) * $(win)- P(lose) * $(lose)} = 0.5*150$+0.5(0.2*300$-0,8*200$) = 25$

In this example, raising all-in is the best case scenario!

Note that in order to make exact calculations you would have to factor in the case that your opponent has a set of Aces or Tens and makes a full-house when you make your flush by hitting the 8♥, or 4♥. On the table, you do not have to make exact calculations. You can treat these cases by adjusting your outs (considering some outs as half for example), or adjusting the probability that you will win the hand.

So, when can Expected Value be employed?

Calculating the expected value has many applications in poker. It can be used on the table when you are facing a relatively simple decision, like to call, fold, or to bet all-in, with all the betting ending after the decision. If there are subsequent betting rounds, the calculations may become difficult, at least when you are on the table. It can also be very valuable to do some off-table calculations and study examples like the one above, to improve your intuition and also improve your capacity to calculate estimations of the expected value when you are on the tables.

In the next tutorial, we will present another tool called pot odds, which can sometimes be used as a shortcut to reduce the computations that are necessary to calculate the expected value.

Please leave any comments or questions that you may have. You can also check out our recommended poker sites page!


This tutorial is part of the Advanced Poker Strategy Course. You can continue to the next tutorial on Implied Odds!

10 thoughts on “What is Expected Value in Poker?

  1. Helo Dear, a big thanks to you for sharing this educative post on poker wand. This is exactly what I have been looking for and it’s really going to be helpful to me. I have been encountering series of challenges in calculating the expected values in poker, But I must say I am lucky seeing this. I have learnt a lot. Thanks

    1. Hello Sheddy,

      If working with Expected Value seems challenging, you can check out the tutorial on Pot Odds. You can use pot odds instead of expected value in many cases and works as a shortcut technique as it requires fewer calculations. I wish you all the best on and off the tables 

  2. Hey nice article you have there. As a newbie in the poker world, I am still learning about some of its strategies, having stumbled into this article, I have learned a whole lot of new tips that will definitely help me in all of my transactions in poker. Before now, I have been wondering how the expected value is calculated and employed, this article has made it much easier for me

  3. So it turns out folding can be a clever move, given the expected value of the other option is going to be negative. I’ve always thought that backing off is a cowardly move to make. Now I’m starting to wonder how to utilize expected value when I play my next game of poker. By the way I love your content here, I guess I’ll be referring back to your site often when it comes to winning at poker. Keep it up!

  4. Hello there, This is an amazing article that you have got here. This is so well written and detailed, I believe any one and every one should find it really easy to understand. I have had little knowledge about poker and I have found this article very helpful to me.I’m so blessed by this write up, hope to see more on your website.

    thanks for sharing this with me.

  5. Hi

    It is a great way of analysing  of what to do with you hand by basing it on a simple mathematical prediction. I can imagine it can get very complicated if you are trying to calculate with 3 or more players still in the hand. You would need a computer program to know if it is viable or not. I still feel that most poker players still rely on their guts more than calculation.

    What do you do if multiple people still remain in the hand?

    Thanks

    Antonio 

    1. Hi,

      With 3 or more players in the hand, and also with more betting rounds to come things get exponentially complicated. This is why poker cannot be resolved to a simple algorithm that tells you what to do at any given time. Still, you can use some basic poker strategies, not rely solely on intuition. 

      For example, you can put your opponents on rough estimations of their hand ranges, and make an approximation of your equity against them. Then you can ponder on how future cards may affect the dynamics, how many cards can help you (your outs), and how many cards can harm you. 

      Finally put it all together to find the best course of action. If you are behind, is what the pot is offering, combined with the implied odds, enough for you to continue to draw your hand? This cannot be calculated by a precise equation at the table but you can use shortcut techniques like pot odds to help you. 

      If you are ahead, how should you bet to maximize value and protect your hand?

      Thanks for your comment

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