Today we’re going to discuss expected value — what it means, why it’s important, and how to apply it. Expected value is one of the most fundamental concepts in poker, and one of the most important to understand.
Table of Contents:
- What is Expected Value (EV)?
- Real World Example: The Steph Curry Bet
- Expected Value in Poker
What is Expected Value (EV) in Poker?
In short, expected value (EV) is the average result of a given play if it were made hundreds (or even thousands) of times.
Let’s start with a non-poker example to understand its basic application, and then we’ll move on to a poker hand example.
The Steph Curry Bet
Suppose you sneak into the Golden State Warriors practice facility, interrupting Steph Curry’s free throw session. Strangely, he doesn’t have you removed by security, but instead proposes a $5 bet on his next shot.
You hesitate. Steph Curry is, of course, an incredible free throw shooter. But he’s getting impatient. “C’mon. It’s only five bucks!” he says.
Should you take the bet?
Your first inclination is probably to decline, and you’d be correct to do so. At even money—your $5 versus Steph’s $5—this bet has a negative expected value (-EV).
Let’s see why that is:
As you can see from the chart, there are two possible outcomes in this scenario:
- Curry makes the free throw and you lose $5, which will happen 90.1%* of the time, or
- He misses and you win $, which will happen 9.9% of the time.
*Curry’s career free throw made percentage is very close to this number.
By looking at the two outcomes and their likelihoods, your can probably tell intuitively that this is a losing bet for you. But exactly how much is it losing?
How to Calculate Expected Value
To calculate the EV of this bet, we simply multiply the probability of each outcome (as a decimal) by its respective result, and add them together (for a less hands-on approach, you can use an expected value calculator):
MISS: 0.099 x $5 = $.495
MAKE: 0.901 x -$5 = -$4.505
MAKE + MISS: -$4.505 + $0.495 = -$4.01
EV = -$4.01
Your intuition was correct. If you made the bet 100 times over, you should expect to lose a lot of money: more than $4 per shot. Ouch!
So, you decline the bet, and Steph scowls.
But now suppose that he offered you 20:1 odds: his $100 to your $5. What should you do?
Let’s recalculate and see if this new bet is worth taking.
MISS: 0.099 x $100 = $9.90
MAKE: 0.901 x -$5 = -$4.505
MAKE + MISS: -$4.505 + $9.90 = +$5.395
Now we’re talking! With the odds Curry has offered you this bet is +EV. So, you accept.
Steph then steps to the line, drills the free throw, and snatches the $5 from your hand.
Perhaps you’re thinking, what?! The bet was supposed to be +EV!
It was +EV, and you ought to make the bet again if given the opportunity. But that doesn’t mean you will always win. In fact, based on what we know about Steph Curry’s free throw shooting percentage, you should expect to lose 90% of the time—he’s just that good at shooting free throws.
The important point is this: it’s only by making the bet often enough that your expected value might be realized.
Expected Value in Poker
It’s easy to see how EV works at the poker table.
To take an easy example, just think of how many times you’ve had pocket aces cracked after going all-in preflop. With very exceptional cases set aside (certain rare bubble and pay jump scenarios in tournaments), would you ever have considered folding those aces in hindsight?
Of course not. Because you know that getting your money in before the flop with pocket aces is a hugely profitable play in the long term.
Being a successful poker player depends on consistently making profitable (+EV) plays, many of which are more difficult to identify than others, and putting in enough volume to overcome negative variance (instances when you make the correct, +EV play, but still lose the pot), which is inevitable.
Let’s consider an example.
EV Example: Should You Shove All-In with a Combo Draw?
Suppose you’re on the button with $200 in a $2/4 full-ring cash game. A loose opponent opens to $16 from early position, and you call with J♦ 9♦. Both blinds elect to fold, leaving you heads up. The pot is $38.
The flop comes 5♦ 10♦ 2♣, and Villain fires a $30 continuation bet. You decide to call, making the pot $98, and leaving you with $154 behind.
The turn brings the 7♠. Villain bets $50. The pot is now $148.
Calling is a reasonable option, but let’s consider the EV of an all-in shove.
Let’s assume you’re familiar with Villain’s game, and know that she’s very capable of putting on the pressure with marginal holdings. You therefore think that if you shove she might fold 66% of the time. On the other hand, if Villain calls, you will need to hit your combo draw to win the pot.
Let’s see if this play is +EV based on the assumption that when Villain calls, it will be with a hand like T9 suited for top pair, against which your draw will have 34.09% equity. We’ll follow the same process as the Steph Curry example, beginning with this decision tree:
There are three possible outcomes as shown on the tree:
- Villain folds and you win $148 (her surrendered $50 plus the $98 pot).
- Villain calls and you miss your draw, which results in you losing $154 (your all-in shove).
- Villain calls and you hit your draw, which results ion you winning $252 (the $98 pot plus her $154).
Calculating the EV for the first outcome is easy:
Villain Folds: $148 x 0.66 = $97.68
Now, let’s calculate the EV when called based on these numbers (remember: when she calls, you’ll either lose $154 or win $252):
Villain Calls and You Lose: 0.6591 x -$154 = -$101.5014
Villain Calls and You Win: 0.3409 x $252 = $85.9068
EV When Called: -$16.5014
Let’s plug that number back into our tree.
Now we can assess this play.
Villain Calls: 0.33 x -$16.50 = -$5.45
Villain Folds: 0.66 x $148 = $97.68
EV of Shove: (-$5.45 + $97.68) = $92.23
Hurray! Shoving is indeed profitable.
But before you snap-shove every flush draw, it’s important to dig deeper into why that’s the case.
One reason why this a good bluff spot is that when called, you’re never drawing dead. You have great equity based on the number of outs you have: any 8 gives you the nut straight, any diamond completes your flush, and if Villain is holding a hand like T9, as we’ve assumed, then a jack will also give you the win.
Based on your read, any of these 15 cards (9 remaining diamonds + 3 non-diamond 8s + 3 Jacks) will win you the pot. With one card to come, this puts your equity—the likelihood that we’ll win a given hand—at 34%. It’s not ideal, but when an opponent won’t be calling often, it’s a great spot to shove.
It’s important to notice how significant the assumption that Villain doesn’t have a good hand is—she’s aggressive and splashes around in a lot of pots, so you’re confident that she is capable of firing two bets with a marginal or weak hand that will fold to a shove.
So long as this read is well founded, your shove is great. But what if you’re wrong? What if the Villain was a very tight player instead? In that case, your EV goes way down, since the likelihood of you shove getting called goes way up.
For example, suppose the Villain is a tight player who would only bet on the turn with strong hands. In this case, the odds of Villain folding diminish almost entirely, meaning you will need to hit your draw to win. Against such an opponent, calling the turn bet is a much better option than shoving.
In other words, this can’t always be a slam dunk shove. It’s only significantly +EV against certain opponents who will actually fold versus your shove at some frequency. It’s up to you to assign ranges to determine when to shove and when to fold.
Variance, The Long Term, and Putting It All Together
If there’s a single crucial take away from this article is that ‘expected value’ is just that: expected, not guaranteed.
It’s not unusual to go on extended cold streaks and run significantly below what EV would dictate. Although it’s largely game of skill, there is an element of luck to poker that you cannot avoid. The only way to mitigate against that is by making +EV plays as often as possible, and by putting in serious volume.
So, how do you apply these lessons to your game? You obviously can’t construct a decision tree for every hand you play, but there are a few tools that can help you while you play, including preflop charts, poker expected value calculators, and poker EV software. If you’re serious about your game, I highly recommend checking them out.
Until next time—play great and run better!