Mathematics: How does Knowing the Expected Value (EV) Help You?
Introduction
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.
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:
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.
How to use the EV to make the best decision
What's looked easy so far can quickly become complicated when you start analyzing real situations. If you happen to find yourself with the choice of folding or calling on the turn, for example, simply do this: Calculate the EV of calling, and then the EV of folding. Compare the results - the option with the highest value is the decision you want to make.
Let's take a look at an abstract example first. You have two events: A and B. The probability that A happens is 80%, the probability of B happening is 20%.
| Event A happens | Event B happens | |
| Probability | 80% | 20% |
In this example, you can bet on whether Event A or B will take place. If you bet correctly on Event A, you win €2, but will lose €4 if Event B takes place.
Or, you can win €4 by correctly betting on Event B, but will lose €2 if Event A takes place.
| Event A happens | Event B happens | |
| You bet on A | You win 2€ | You lose 4€ |
| You bet on B | You lose 2€ | You win 4€ |
To find out whether it's better to bet on A or B, you compute the expected value for a bet on Event A and the expected value for a bet on Event B. Then you compare them.
EV = (possible winnings) * (probability of winning) - (possible losses) * (probability of losing)
EV(A) = 2€ * 80% - 4€ * 20%
EV(B) = 4€ * 20% - 2€ * 80%
The EV of Event A results from the fact that you will win €2 80% of the time and lose €4 20% of the time. Let's find out exactly what that means.
EV(A) = 2€ * 80% - 4€ * 20%
EV(A) = 1.6€ - 0.8€
EV(A) = 0.8€
The resulting value is positive, meaning it is profitable to bet on Event A. You will win €0.80 on average every time you do. Now let's take a look at the EV of Event B.
EV(B) = 4€ * 20% - 2€ * 80%
EV(B) = 0.8€ - 1.6€
EV(B) = -0.8€
The resulting value is negative. You will lose €0.80 on average every time you bet on Event B!
You compare the EV of both possible decisions and see that the EV of A is significantly higher than the EV of B.
EV(A)>EV(B)
A win of 80 cent on the one side, and a 80 cent loss on the other side. Your best choice will obviously be to bet on A. And just like that, you have learned an essential principle in poker.
This is a fundamental principle of poker. Take a closer look at the wording. You aren't looking for the action that will result in the highest profit, but rather for the action with the highest expected value. Sometimes the best action will be -EV, meaning you will lose money on average. This is still the best decision when all other options have an even lower EV.
Expected value in Texas Hold'em
Suppose you are facing one opponent on the turn.
You have: T
9
The board shows: 6
7 2 4
You have a gutshot straight draw and need an eight to make your straight. You are certain that your opponent has an overpair, like two queens or maybe kings.
Your outs are clean and you need one of the four eights in the deck to win the hand. 4 of the 46 cards remaining in the deck are useful. The probability of hitting an 8 on the river is, therefore, 4/46, or about 0.087 or 8.7%.
You lose when any other card is dealt. There are 46 - 4 = 42 cards in the deck that are not useful to you. The probability of losing the hand is, therefore, 42/46, or 0.913 (91.3%).
All we need now is the possible winnings and losses. The pot is $5. Your opponent bets $1. This means the pot you can win is $6 now.
Your losses are the $1 you have to pay to call.
The EV of calling is therefore:
EV(call) = (possible winnings) * (probability of winning) - (possible losses) * (probability of losing)
EV(call) = $6 * 8.7% - $1 * 91.3%
EV(call) = - $0.39
A negative value, meaning you will lose money on average. You can expect to lose $0.39 of every $1 you invest, and win $0.61 of every $1 you invest. A poor investment and a clear case of -EV.
But don't stop there. You know how much you can expect to lose on average by calling, but you need to determine the EV of all possible actions before you know which decision is best.
The EV of folding is always zero.
EV(fold) = $0
You don't invest any more money, meaning your possible losses are zero. The same goes for your winnings though, as you can't win the hand when you fold. This is where a lot of people ask, "What about the money you've already invested in the hand?" The answer: Forget that money, it's not yours anymore.
Compare the EV of all possible actions to determine the correct decision:
EV(fold) = $0
EV (call) = -$0.39
EV(fold) > EV(call)
You can clearly see that you are better off folding than calling.
Other uses
Playing good poker means making the decision with the highest EV in a given situation. Any time the expected value of an action is higher than 0, you make profit by taking that action.
Once you start playing around with the formula (which you should do when analyzing past hands where you weren't sure about your choice, for example) you can start asking yourself questions like: How big would the pot have had to be in order to justify calling a $1 bet with my gutshot draw? Let's take a look.
The pot size is therefore our "possible winnings":
EV(call) = (possible winnings) * (probability of winning) - (possible losses) * (probability of losing)
You want to know when the expected value will positive:
EV(call) > 0
We can change the formula to find out how large the pot must be for the EV to be positive:
The EV of calling is > 0, when:
(possible winnings) * (probability of winning) - (possible losses) * (probability of losing) > 0
Now we just have juggle a few things around. You might want to brush up on your algebra if you have trouble following this.
The EV(call) is > 0, when:
(possible winnings) * (probability of winning) - (possible losses) * (probability of losing) > 0, which can be rewritten as:
(possible winnings) * (probability of winning) > (possible losses) * (probability of losing), which can be rewritten as:
(possible winnings) > (possible losses) * (probability of losing) / (probability of winning)
Now we have isolated our unknown variable on the left side of the formula and can plug in the values that we have to arrive at the answer!
(possible losses) = $1
(probability of losing) = 42/46
(probability of winning) = 4/46
This leads us to the following result:
(possible winnings) > (possible losses) * (probability of losing) / (probability of winning)
(possible winnings) > $1 * (42/46) / (4/46)
(possible winnings) > $1 * 1932/184
(possible winnings) > $10.5
This means the pot has to be bigger than $10.50 for a $1 investment to be profitable. You should already know the odds for a gutshot straight draw on the turn are 1:11. You can therefore afford to call when you can win at least 11x that amount.
The method of expected value has brought you to the same result as you would have arrived at by calculating the odds. There is, in fact, a very simple connection between the two. As we said, the odds of hitting one of your four outs for a straight are 4/46, which equates to 0.087, or 8.7%.
In other words: You will complete your straight app. 1 in 12 times (100% / 8.7%). You therefore have 11:1 odds against you, meaning you will miss 11 times and hit once every 12 times in this scenario. As you can see, the EV is directly translateable into the odds format, in which pot odds are also generally given.
It's the same idea, just a different way of getting there.
Conclusion
Let's review. To compute the expected value, you weigh your possible winnings against your possible losses.
EV = (possible winnings) * (probability of winning) - (possible losses) * (probability of losing)
This tells you how much you will win or lose in average.
You can simply define winnings and losses as payout and use this simpler formula:
EV = Probability1 * Payout1 + Probability2 * Payout2 + ... + Probabilityn * Payoutn
A positive result means you will make a profit, a negative result means you will suffer losses.
When you analyze a situation, you have take all possible results, as well as the probability of that result taking place, and how much you will win/lose into consideration. Then you put these values into the formula to determine the EV of a given action.
If you are confronted with several possible choices and want to know which is the most profitable, you compute the expected value for every possible action. Then compare the values and choose the course of action with the highest EV.
You've now learned how to take your game to a new mathematical level by determining the EV of a decision in a given situation. All this calculating might seem complicated at first, but it's not really all that complicated once you get the hang of it. Don't hesitate to ask for help in the forum if you're having trouble.
A lot of articles and concepts you will come across in the future require you to calculate the EV of all possible decisions. Take the time to master the concept of EV. Understanding the concept of expected value is the next step after mastering the concepts of odds and outs.
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