We've explained what EV is in another place. Here, we'll show you exactly how EV is calculated.
There is going to be some math. Actually, the whole purpose of this article is to explain the math behind the EV. So, fasten your seat belts.
Uncertain outcomes
You're in a situation where a number of things can happen - you just don't know which one it will be. So there are a number of possible outcomes.
Each outcome is more or less probable, and each one represents a certain value to you - such as winning or losing a number of chips in a game of poker.
Notation
The situation is formalized like this:
- There are a number of possible outcomes, A1, A2, A3...
- Each outcome has a probability of occurring: P(A1), P(A2), P(A3)...
- Each outcome has a value for you: V(A1), V(A2), V(A3)...
The EV of this situation is calculated as follows:
EV = P(A1)*V(A1) + P(A2)*V(A2) + P(A3)*V(A3) + ...
In short, you multiply the probability and value for each outcome and sum them up.
What you get is the EV - the expected value or expectation value.
It's the value that this situation represents to you, all in all, given its uncertain outcome.
Example: You're all in on the turn with a 60% chance of winning. There's $100 in the pot.
Two outcomes are possible (assuming you cannot get identical hands and split the pot):
A1: You win. P(A1) = 0.6. V(A1) = 100
A2: You lose. P(A2) = 0.4. V(A2) = 0
Your EV is $60:
EV = 0.6*100 + 0.4*0 = 60
Note: This EV is sometimes referred to as Sklansky dollars - the amount you will win or lose - in the long run.
Example: You and a friend bet $10 each on a dice roll. If you roll 7 or better you win; if not, you lose.
A1: You roll > 7. P(A1) = 21/36. V(A1) = 10
A2: You roll < 7. P(A2) = 15/36. V(A2) = - 10
EV = 10*21/36 - 10*15/36 = 60/36 = 1.67
You're expected to win $1.67.
If you play once, you'll either win $10 or lose $10. But if you play this game many times, your average profit will approach $1.67.
Read more about Average, EV, Variance and More.
/Charlie River
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This article is part of the poker math series:
- Basic Probability Theory
- Where do Probabilities Live?
- Average, Expected Value, Variance and More
- What Is Odds?
- What is Outs in Poker?
- The Math behind Calling and Folding
- EV - How To Calculate It
- Variance - How to Calculate It
Comments on this Article
Charlie River (Oct 11, 2010)
No, S, that would be the total profit.
Maybe I should have said "you average profit per game will appraoch $1.67"
Basically, only ratios have this nive behavior. Sums tend to get further and further away from the expected value, in a statistical sense.
s (Oct 08, 2010)
" But if you play this game many times, your average profit will approach $1.67."
If I'm right this is : " But if you play this game many times, your average profit will approach $1.67 x n where n is number of tries ."
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