Mark  Poker Articles, Poker Mathematics
This article belongs to the Poker Mathematics series.
Before going into depths with the concept of EV in poker, some math is needed. In mathematics, the EV or expected value of a variable is defined as the probability weighted sum of the possible values for the variable in question. This may sound complicated, but it really isn’t that bad. Let’s look at the simple example of rolling a dice. In this case the possible values and probabilities are as follows:
 1 will come up in 1 out of 6 rolls on average yielding a probability of 1/6 = 16,7%
 2; 16,7%
 3; 16,7%
 4; 16,7%
 5; 16,7%
 6; 16,7%
So the EV for rolling a dice is: (1 x 0,167) + (2 x 0,167) + (3 x 0,167) and so on. In total this adds up to 3,5.
When it comes to EV in poker, there are basically two situations to consider, namely EV when you call a bet and EV when you make a bet.
In both cases, I will define EV in poker as the amount you will be payed back on average on a 1$ bet (including your original 1$ bet):
 EV < 1 – If you always play hands with an EV of less than 1 you will lose on average
 EV = 1 – If you always play hands with an EV equal to 1 you will break even on average
 EV > 1 – If you always play hands with an EV of more than 1 you will win on average
So ALWAYS aim to play hands with an EV larger than 1!!!
EV IN POKER WHEN CALLING A BET
It is pretty straight forward to calculate your EV when calling a bet. It’s simply the product of your probability (P) of winning the hand and the decimal pot odds (DPO) offered:
 EV = P*DPO
For a review of the difference between decimal odds (European style odds) and fractional odds (American style odds) check out my article: poker pot odds; all you need to know.
For an easy rule of thumb to calculate poker probabilities check out my article: poker probabilities; all you need to know
At this point I think an example is called for (warning, there is some math content ahead):
Let’s say you are playing an online tournament and the pot on the turn is 3000. Your only remaining opponent bets 500, what is your EV for a call? With your opponent’s bet of 500 the pot is now 3500 and you have to call 500 to stay in the pot.
In order to calculate your decimal pot odds, simply add the amount you have to call (in this case 500) to the pot (3500+500) and divide by the amount you have to call (4000/500 = 8). So your decimal pot odds is 8 in this case.
Let’s also assume that you have a total of 12 outs to win the hand on the river. Using the easy rule of thumb to calculate poker probabilities, this means that your probability of winning the hand come showdown is (2*12+1)% = 25% or alternatively 0,25.
For simplicity we are assuming that the river is checked by both you and your opponent so no extra chips enter the pot.
Now all that’s left to calculate your EV in the situation above is to multiply the probability with the pot odds, yielding an EV of 8*0,25 = 2.
This means that for every chip you call in the situation above, you will on average get 2 chips back. Or as in the case above with your 500 chip call, you can expect to get 1000 chips back, meaning you win 500 chips in total.
This result fits very well with the mathematical definition I gave of the expected value in the beginning of this article. Say you make the 500 chip call 100 times. In 75% of the cases you will lose your 500 chips, corresponding to 37500 chips. In 25% of the cases you will win 3500 chips, corresponding to 87500 chips. This means you will win on average 50000 chips in total over 100 plays, or 500 chips per play which fits perfectly with our previous result.
EV IN POKER WHEN MAKING A BET
EV in poker when making a bet is all about pot and pot odds control. You want to be choosing your bet sizes so your opponent’s EV for calling is less than 1.
Say you have a high pair and the flop comes with 2 hearts. The pot has 2000 chips in it and you’re up against 1 opponent as the first to act. In order to make sure your opponent’s expected value for calling is less than 1, you need to pay attention to the size of the bet you make. If your opponent is on a flush draw he has 9 outs to make his flush on the turn providing you do not have any hearts in your hand. 9 outs correspond to a probability of roughly 20%. What size bet should you make?
In order to answer this we need to revisit the formula for calculating EV in poker:
 EV = P*DPO
An EV equal to 1 means that you will break even on average. Inserting this into the formula above we get the following relationships between the probability P and the decimal pot odds DPO:
 DPO = 1/P
 P = 1/DPO
So for a probability of 20% for your opponent to hit his hand on the turn, you need to be offering him pot odds 5 (1/0,2) for him to break even on average and less than pot odds 5 to ensure he has an EV less than 1.
If you bet half the 2000 chip pot in the example above your opponent will be getting pot odds 4 to call, corresponding to an EV of 0,8 meaning that he on average will lose 0,2 chips for every chip he bets.
I hope this article has made you realize that understanding the concept of EV in poker is key factor of becoming a winning player in the long run.
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2 Comments to EV in poker
That doesn’t works out always. You should also consider other nonmathematical factors…
March 31, 2010
Hi Vivek91
Thanks for your comment. What nonmathematical factors do you mean? Can you elaborate?
Leave a comment
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March 29, 2010