The phrase "expected value" is a phrase that didn't enter the sphere of poker jargon until relatively recently. Low stakes regulars before the poker boom were certainly not discussing it, but these days it's tough to get too far into a respectable poker book or strategy subforum without seeing it loosely tossed around. In essence, "expected value" is a concept in probability that is used to describe the average outcome of a given scenario when the scenario hinges on an uncertain, probabilistic event.
Let's take a standard, unweighted coin as an example. A fair coin, when flipped, has a 50% chance of coming up heads, and a 50% chance of coming up tails. I tell you that I'm going to flip the coin, and pay you $1 if it comes up heads, while paying you nothing if it comes up tails. It's clear that there is a 50% chance of me paying you a dollar and a 50% chance of me paying you nothing. The average outcome here is also fairly straightforward. You would expect, on average, to be paid 50 cents.
But there's a little bit of a disconnect here. The 50 cents answer seems obvious here, but how did we actually get to it? It feels somewhat unintuitive because we don't ever actually get paid 50 cents. We are either given $1, or $0. The expected value in this example is described by considering what happens when we take the scenario and repeat it many, many times - an infinite number of times in fact. We have to do this because it takes an infinite number of flips for us to be guaranteed that we flip an equal number of heads and tails.
Given that our probabilities will "work out" to be correct in the long run, we can use them to calculate the average outcome for our scenario, which inherently assumes the long run. To do this, we take the probability of each potential outcome (heads or tails), and multiply it by its corresponding payout ($1 or $0). (For the following equations, P(A) will denote the probability of event A happening, and amt(A) will denote the payout that occurs when event A happens.)
EV(coin flip) = P(heads)amt(heads) + P(tails)amt(tails)
EV(coin flip) = (50%)($1) + (50%)($0)
EV(coin flip) = $0.50 + $0
EV(coin flip) = $0.50
So we see that our intuition here served us well - the expected value is indeed 50 cents. But what happens if we change the scenario? I tell you that if I flip heads, I will pay you $1, while if I flip tails, I will pay you $3. Do you know what the expected value is? It's likely you can still do this in your head, or that you figured out that the answer will be halfway between the two outcomes. Let's quickly apply the formula above to spot-check ourselves:
EV(coin flip) = P(heads)amt(heads) + P(tails)amt(tails)
EV(coin flip) = (50%)($1) + (50%)($3)
EV(coin flip) = $0.50 + $1.50
EV(coin flip) = $2
You may be wondering why we even bother calculating expected value if it's so easy to do in our heads. But there are plenty of other scenarios where we can't calculate this in our heads. One last coin flip example:
I tell you that I will pay you $3.80 if you flip heads, and you will have to pay me $2.60 if you flip tails. Additionally, the coin is weighted toward tails, and will come up 70% of the time. Should you accept the offer? Let's go back to our formula (keeping in mind that if you lose, your payout is a negative value):
EV(coin flip) = P(heads)amt(heads) + P(tails)amt(tails)
EV(coin flip) = (30%)($3.80) + (70%)(-$2.60)
EV(coin flip) = $1.14 - $1.82
EV(coin flip) = -$0.68
I bet you didn't know that one off the top of your head - and there are a ton of applications of the concept of expected value to poker.
Limit hold’em
In poker, expected value still defines the average outcome when taking into account the probability of certain events. With so many situations where probability dictates the outcome, it stands to reason that the concept of expected value will apply a lot in poker situations. To illustrate this, let's look at a simple limit hold'em example:
Your opponent has checked, and it's your option. The pot is $100, and the big blind is $10. Should we bet? To answer this, we'll first have to determine how often he calls, how often he raises, and how we fare against each of those. Let's assume that we somehow know his exact range and it is only Ax, where the non-ace card is a club. We think that A9, A7, and A6 will call, and A5, A4, and A2 will fold. AQ with the Qc or AK with the Kc will make it two betting rounds. Clearly, if he makes it 2 bets, we should always fold, since we are always losing. But should we bet in the first place? Let's look at our EV formula, and state that situation X is the one where we bet, he raises, and we fold. Situation Y is where we bet and he calls. Situation Z is where we bet and he folds.
EV(bet) = P(X)amt(X) + P(Y)amt(Y) + P(Z)amt(Z)
We know that each of his possible hands are equally likely (2 aces left, times the specific other card means two combos each), and there are 8 total hands he can have. So we can quickly determine the probability that he holds each type of hand.
P(X) = 2/8 = 0.25
P(Y) = 3/8 = 0.375
P(Z) = 3/8 = 0.375
When we bet and get raised, we have lost our $10 bet. When we bet and get called, we make the $100 in the pot, plus the $10 from the call. When we bet and he folds, we just win the $100 in the middle.
amt(X) = -$10
amt(Y) = $110
amt(Z) = $100
EV(bet) = (0.25)(-$10) + (0.375)($110) + (0.375)($100)
EV(bet) = -$2.50 + $41.25 + $37.50
EV(bet) = $76.25
This seems great right away - but to know whether it's better than checking, we must also determine the EV of checking. The probability is a little different this time, because there are only two outcomes - we win at showdown, or lose.
P(win) = 6/8 = 0.75
P(lose) = 2/8 = 0.25
The payouts are different too. When we win, we win the $100 in the pot, but when we lose, we don't lose anything additional.
amt(win) = $100
amt(lose) = $0
So putting this all back together, we can find the EV of checking:
EV(check) = P(win)amt(win) + P(lose)amt(lose)
EV(check) = (0.75)($100) + (0.25)($0)
EV(check) = $75
We find that this is actually fairly close. The difference in EV of betting as compared to checking is only $1.25, or 1/8 of a big bet. This is not a slam-dunk, huge difference, so if we have some uncertainty in our reads about his range, it could be a spot where a check would be preferable. For instance, if he actually raises the 9c but we don't think he does, we may fold incorrectly and lose the pot too often, decreasing the EV of a bet below the EV of a check.
No limit hold’em
Let's take a look at an example from no limit hold'em. We're on the river and we hold J9dd on a board of Ad Kd 4c 7h 8s, for nothing but a busted flush draw. Our opponent has checked to us, and we are deciding whether or not to bluff all in. The pot is $200, and we have $100 in our stack.
Due to how the hand played out, we think our opponent can't have sets or flopped two pair, but he could hold AQ-AT, QTdd, QQ-99. We think that if we shove, he will fold QTdd, QQ through 99, but will call with his Ax hands. Let's look at the number of each of those hands, and the resulting probability that he will call a shove:
AQ-AT = 36 combinations
QTdd = 1 combination
QQ-99 = 18 combinations (we block Jd and 9d)
Total = 55 combinations
P(call) = 36/55= 0.6545
P(fold) = 19/55 = 0.3455
If we shove and he folds, we win the $200. If we shove and he calls, we are always beat and we lose our $100 stack.
amt(call) = -$100
amt(fold) = $200
EV(shove) = P(call)amt(call) v P(fold)amt(fold)
EV(shove) = (0.6545)(-$100) v (0.3455)($200)
EV(shove) = -$65.45 v $69.18
EV(shove) = $3.73 difference
The expected value of a check is actually $0 here, because we never win the pot when we check it behind.
EV(check) = $0
We see that shoving is slightly preferable to checking, even though we get called well over 50% of the time. This is a situation that can arise fairly often in poker - since most bets are for a pot-sized bet or less even in no limit games, you are typically laying yourself a price of better than 1:1. In this case we lay ourselves 2:1 by risking a bet worth half the pot, so we only need it to succeed 1/3 of the time to be breakeven as compared to checking.
There are a lot of deep, fundamental applications of expected value in poker, many of which won't help you too much at the table. However, expected value does have application in real-time poker situations. This is usually going to be in special cases where either
Anything beyond these three tends to get vastly more complicated as you start to add more branches to the possible decision tree. It's still possible to do these types of calculations using software, but in-game they are nearly impossible. However, with some practice it's possible to gain a level of proficiency where you can do rough EV estimates for fold equity situations and river value bet spots. This requires a deep understanding of your equity, as well as the ability to do basic algebra on the fly, but estimates will usually get you pretty close. The true power of expected value comes in doing off-table calculations that help you improve your intuition and get your in-game estimates closer and closer to the real expected values of various plays.
Continue learning odds with our next guide in this series, on Pot Odds & Implied Odds. For more information on the math of poker, visit Wolfram MathWorld. Also, start using our Poker Odds Calculator to help figure your odds on any given hand.