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Expected Value

Expected value, or EV, is at the heart of poker math.  EV is the average win or loss of a specific situation if one was able to repeat the exact situation an infinite number of times.  Poker EV is simply the application of expected value to poker situations.  

Expected Value Formula

To calculate expected value, the outcomes of a situation and their respective probabilities are applied to the expected value formula.  The formula consists of two simple steps:

  1. Multiply each possible outcome by the probability of that outcome occurring.
  2. Add the products together-this the EV of the given situation.

Expected Value Formula for Poker Players

Expected Value Examples

 A bet can be positive, negative, or neutral EV.  Let's say there is $100 in the pot on the river with the board reading 9♠8♣2♦K♥3♥ and our lone opponent goes all-in.  

For all 3 examples, we are holding Q♠Q♥ and while deciding whether or not to call, our opponent tells us that he either flopped a set or he missed an open-ended straight draw holding specifically JTs.  Therefore, he can have 9 sets (3 combinations of each flop card) and 4 combinations of bluffs (the 4 ways to make JTs).

In example 1 our opponent has only $50.  In example 2 he has $80, and in example 3 he has $100.  The following charts illustrate what would happen if the hand played out 13 times.  

Example 1:  $50 stack size

Trial

Outcome

Win/Loss

1

Win

150

2

Win

150

3

Win

150

4

Win

150

5

Lose

-50

6

Lose

-50

7

Lose

-50

8

Lose

-50

9

Lose

-50

10

Lose

-50

11

Lose

-50

12

Lose

-50

13

Lose

-50

Total Profit

150

Profit/hand

11.54

Here we would win $150 the 4 times our opponent has the busted straight draw, but lose $50 the 9 times he has a set.  After 13 hands we profit $150 total or $11.54 per hand.

Example 2:  $80 stack size

Trial

Outcome

Win/Loss

1

Win

180

2

Win

180

3

Win

180

4

Win

180

5

Lose

-80

6

Lose

-80

7

Lose

-80

8

Lose

-80

9

Lose

-80

10

Lose

-80

11

Lose

-80

12

Lose

-80

13

Lose

-80

Total Profit

0

Profit/hand

0

Here we win $180 the 4 times our opponent has the busted straight draw and lose $80 the 9 times he has sets.  After 13 hands we have broken exactly even.

Example 3:  $100 stack size

Trial

Outcome

Win/Loss

1

Win

200

2

Win

200

3

Win

200

4

Win

200

5

Lose

-100

6

Lose

-100

7

Lose

-100

8

Lose

-100

9

Lose

-100

10

Lose

-100

11

Lose

-100

12

Lose

-100

13

Lose

-100

Total Profit

-100

Profit/hand

-7.69

Here we win $200 the 4 times our opponent has a busted straight draw, but lose $100 the 9 times he has sets.  After 13 hands we have lost $100 for a net loss of $7.69 per hand.  

Takeaways

Despite winning and losing outcomes being possible in all 3 examples, they are very different bets in the long run.  

We want to make the bet in example 1 as often as possible, are indifferent to the bet in example 2, and want to avoid the bet in example 3.  Why?

This scenario has been simplified and each possible outcome charted at their actual frequencies.  However, because a deck of cards is random, our opponent may show up with a set 20 times in a row in example 1.  This bet is still positive EV despite it being a loser over those 20 hands.  As long as we keep making that bet, we will win.  As the number of trials gets larger, the actual results will approach expected results.  This is why it is important to consistently make positive EV decisions.  

Display odds of winning in all-in pots at Paddy Power Poker.

EV Graphs

When results are poor in the short run, poker players often use all-in EV graphs to evaluate their play.  These graphs show actual results vs expected results.  This can be helpful to determine if a player is just getting unlucky, or "running bad" over a short sample of hands.  

An example of an EV graph is shown below.  The green line indicates the actual amount won, while the yellow line indicates the expected amount won.

Example EV Graph

Use of these graphs is limited to all-in situations since it is not possible to track the Expected Value of many poker decisions.  This is because many of these decisions involve subjective range analysis so true EV cannot be calculated in real time for every decision made.  

EV Calculations

It is important to know how to find expected value so it can be used to make better poker decisions in the future.  

All-in Example

The simplest poker EV calculations occur when two players get all in with cards to come.  

For example, its a $1/$2 game and all players have $100 stacks.  It folds to the small blind who raises to $10 with T♥T♦.  The big blind now moves all in for $100 holding A♠K♣ and the small blind calls.  What is the small blind's EV of calling the big blinds shove?  

After the big blind shoves, the pot is $110 and it costs the small blind $90 to call.  Pocket tens is an approximately 57-43 favorite over ace-king off suit when all-in preflop.  Therefore 57% of the time the small blind will win $110, and 43% of the time he will lose $90.  The calculation is as follows:

EV= (.57)(110)+(.43)(-90)=  24

This means that on average, the small blind can expect to win +$24 from this situation.  Because this number is positive, he should call.

Bluff Example

It is also possible to calculate the expected value when not dealing with all-in situations.  

For example, on the river in a heads up pot between player 1 and player 2 the pot is $100.  Player 1 bets $50 and player 2 moves all in for $200 total with a complete bluff.  If player 1 calls, player 2 has no chance of winning.  If player 1 folds 70% of the time to the shove, what is player 2's EV of bluffing all-in? 

When the bluff succeeds, player 2 wins the $100 in the pot plus player 1's $50 bet, for a total of $150.  When it fails, player 2 loses $200.  The calculation is as follows:

EV= (150)(.7)+(-200)(.3)= 45

This means that on average player 2 can expect to win +$45 by bluffing, thus he should bluff.  However, if player 1 only folded 40% of the time, the EV would be -$60 and player 2 should not bluff.  

These two basic types of EV calculations can be combined to calculate the expected value of more complicated situations, such as semi-bluffing, in which a bluffer can win when his opponent folds but can also sometimes win when called.

Long Run EV and Poker

Using the above formula, expected value can be calculated for countless poker hands and bets.  

  • A bet that will make money on average is called "positive EV"
  • A bet that will lose money on average is called "negative EV"
  • A bet that breaks even on average is called "neutral EV"

The goal of winning poker players is to make the highest EV bet in each situation they are presented with.  Those that can do this will make money in the long run.  However, because poker is a game that involves variance, it is possible for a positive EV bet to lose money in a specific trial.  Likewise, it is possible for a negative EV bet to win money in a specific trial.  

Good poker players emphasize making the highest EV decisions and do not fret over the outcome of individual trials.  They know as long as they consistently make good decisions they will win, and if they try to get lucky and win money on negative EV bets, they are sure to lose in the long run.  It is important to note that sometimes making the highest EV decision may mean taking the lowest of a series of negative EV bets.  

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