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StrategyFixed Limit

Mathematics: How does Knowing the Expected Value (EV) Help You?


In this article
  • EV = (possible winnings) * (probability of winning) - (possible losses) * (probability of losing)
  • How to figure out the best action
  • How to apply the EV formula to improve your game

The expected value of an action tells you how big your profit or your losses will be on average for that action. In poker, you can always find out which is the "right" action by determining the EV of all possible actions and choosing the one with the highest expected value.

If you are confronted with the choice between calling or folding, for instance, you can compute the EV to know exactly which decision is the better one. Of course, you have already learned how to make a decision regarding call/fold using the odds and outs; learning to calculate the EV is the next step.

In this article you will learn how to calculate the EV of an action and choose the best possible action in a given situation. Furthermore, we will show you how you can use the expected value to analyze past game situations, allowing you to answer questions such as 'What would have had to be the case in order to ...' or 'What would have happened if I had done this?'.

In order to be able to comprehend the content of this article, it is absolutely necessary to understand the concept of outs and odds. You may also want to freshen up on some your basic math regarding probability and linear equations.

How to compute the expected value of an action

Let's suppose someone offers you one of two envelopes. One contains €5, the other €20. You can look and touch, but can't tell which envelope contains which bill.

You can buy one of these two envelopes for €10. The question is, should you? The answer can be found in the expected value of each action. The EV tells you how much money you will make/lose on average if you were to allowed to make this decision repeatedly.

You need four values to compute it:

  • How much can you lose?
  • How much can you win?
  • How high is the probability of winning?
  • How high is the probability of losing?

EV is the abbreviation for expected value, and there is a basic formula to find it.

EV = (possible winnings) * (probability of winning) - (possible losses) * (probability of losing)

You basically measure up possible winnings against possible losses. This gives you the expected value for your decision: What will this decision ultimately leave me with? We see this formula differentiate between winnings and losses, however, we can generally call both of them 'payout'. If the payout is positive, you make a profit, if it is negative, you suffer losses.

Our general formula is therefore:

EV = Probability1 * Payout1 + Probability2 * Payout2 + ... + Probabilityn * Payoutn

Here is what this formula takes into account:

  • What can happen?
  • Payoutx: How much do you earn or lose if event x happens?
  • Probabilityx: How probable is it that x happens?

Instead of differentiating between winnings and losses, you simply determine the result (payout) of a given decision. If the value is positive, you will make a profit on average by making that decision. If it is negative, you will lose money on average by making that decision.

You can calculate an exact EV by taking all possible outcomes in a given situation into account. First, you ask yourself what can happen and how much you would win or lose in each case. After that, you determine the likelihood of each possible result actually taking place. Then you multiply your possible winnings and losses by the probability of that result taking place, and, finally, add up the results.

In our example, the probability of choosing either envelope is 50%. You will take envelope A with €5 50% of the time, and you will choose envelope B with €20 the other 50% if the time.

This also means you will win €5 half the time, and win €20 the other half of the time. However, you will also lose €10 100% of the time you play.

Let's put this into our formula:

  • You lose 10 Euro 100% of the time, since you have to pay to play.
  • You win €5 50% of the time.
  • You win €20 50% of the time.

This translates into the following:

EV = win(envelope A) * 50% + win(envelope B) * 50% – loss(stake) * 100%
EV = 5 Euro * 50% + 20 Euro * 50 % - 10 Euro * 100%

In the next step we see 50% of 5 (€2.50) + 50% of 20 (€10) - 100% of 10 (€10). The result tells how much we would make/lose on average by accepting the stranger's offer and purchasing an envelope.

EV = 2.50 Euro + 10 Euro – 10 Euro
EV = 2.50 Euro

This means you will make a profit of €2.50 on average every time you accept the offer. The result is a positive value, meaning that the decision is +EV. You will hear a lot about 'plus EV' and 'minus EV' in poker and the PokerStrategy community. Now you know what it means.


That's not the entire article...

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Comments (4)

#1 Fongie, 12 Mar 09 11:30

This article should be on the NLHE list aswell, I've been looking for an article concerning EV calculations!

#2 Helipacter, 08 Sep 10 11:06

This is the clearest explanation of EV calculation I have found - it's particularly useful because it shows how you do it inreal game situations! (A supplmentary quiz would be good though!)

#3 datsmahname, 04 Dec 10 07:58

I agree. A quiz on this topic would be an excellent idea. Also, examples including a decision about whether or not we should raise would be nice.

#4 helmyouth, 08 Jun 11 02:01

Any short forms to calculate faster?