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How to Use Expected Value in Poker

Depending how proficient you are in math and in stats, expected value may sound trivial or scary. But it is an absolute necessity to understand EV.

Expected Value or EV in short is a term belonging to the class of statistics, and as poker is primarily a statistical game, EV has found its place in poker. Some players believe that poker is primarily a psychological game, but that debate is not the subject of this article.

Expected Value is not a difficult concept anyway. It is simply the average of all possible outcomes, weighted by their chance of occurrence.

EV can become a bit more convoluted if for example you include the possibility of bluffing or the effect of slow playing. But most pros routinely do it in their head, it is just a matter of habit.

Expected value example

Let us take the simple example of throwing two dice. You bet $1 to play, and if you make a sum of ten, you get $5. What is the EV of this game?

There are 36 ways to throw 2 dice. To get ten, you need {5,5}, (6,4} or (4,6}, 3 ways. So the chance of winning $5 is 3/36 and the chance to lose $1 is 33/36. The EV is $5*(3/36) -$1*(33/36)= ($5 * 8.3%) - ($1 * 91.7%) = $0.42-$0.92 = -$0.50. This is a losing game based on the EV.

It is the exact same idea used to make poker EV calculations, just a little bit more complicated. And if you can calculate precisely the EV of every hand you play, they you have a very mathematical game type like Chris Ferguson who won the WSOP main event in 2000.

In plain English, EV is how much you make if you win times the chance to win, minus how much you lose if you lose times the chance of losing. So the key is to calculate these probabilities of winning or losing, and to apply the formula.

Say you have AK in a $2/$4 heads-up NLHE match with 100 BB stacks, i.e. $400. You raise 3 BB to $12 and get called by QQ. The flop comes Q62, offering you the nuts flush draw versus your opponent's flopped set.

You make an 80% pot c-bet at $20 and get called. The turn is 8, you make another 80% pot bet at $50 and got raised to $200. What is the EV of calling this raise?

There is already $314 in the pot (amount you may win) and you need to add $150 (amount you may lose) to call. Chance of winning: you need a diamond on the river, 9 are left in the deck of 46 cards, 9/46 = 19.6%.

EV = ($314 * 19.6%) - ($150 * 80.4%) = -$59 and you should fold the hand.

In the example above, we assume that we know our opponent's hand. In real games, you do not and you must use a so-called "range" instead of just a fixed hand. What this means is that you need to assess the potential hands that the other player has, and redo the EV calculation with this range.

This is where the playing style of the other guy comes into the equation. Let's say that he is a tight aggressive reasonably ABC straightforward player.

He only called preflop, so {QQ-77,AQs-A9s,AQ-AT,KQs} is his assumed range (rather wide). The only way to calculate the chance of winning using a range is to use an odds calculator such as pokerstove.

Pokerstove gives you a 55% chance to win the hand, as you have more outs plus you dominate many aces in that range. Plugging this number in the formula gives EV = ($314 * 55%) - ($150 * 45%) = +$105.

Ok, this range is probably too wide, because it is unlikely that he would raise 4 times with air. Let us try a tighter range {QQ,88,66,AQ}. Pokerstove gives 21.2% chance of winning which is very similar to the first calculation.

Use your real hand histories and pokerstove to evaluate the EV of your critical hands. After a while the results will come naturally to you and this new poker knowledge will boost your poker skills and bankroll to the next level.


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