Expected Value in Poker
How the Concepts of Expectation and Expected Value Can Help You Ride Out the Poker Swings
Poker is a game where you can make the correct decision, only to see a card hit the river that causes you to lose a big pot. This chance element makes the game exciting, it can also be frustrating at times. Therefore, expected value in poker is a great concept to learn.
Expectation is a key poker concept. It describes how your outcomes pan out when you repeat a play thousands of times. This is long enough for variance in results to even out – as long as you make plays which are profitable over the long-term, the money will eventually come your way. Another way of describing expectation is the term ‘expected value’. This allows you to put numbers on a play. Your moves will have positive or negative expected value. As long as you focus on your game, ensuring that you stick to ‘plus ev’ plays, the money will inevitably come your way.
This page explains the concepts of expectation and expected value in poker, and then goes on to show how it applies to many poker situations. Here is what you will find below:
- Expectation Explained: Simple coin-flip games show expectation in action
- Applying Expectation to Poker: Here I use simple all-in or fold situations to show how this works in poker.
- Expectation with More Cards to Come: How to combine pot odds and outs to figure out if a play has a positive expectation
- Pre-Flop Ranges: This section explains why you should stay tight from early position, and raise more on the button using expected value as a guide
- The Long Run: How to use expectation to figure out your profitability and hourly rates
- Expectation with Winning Players and Fish: Here you’ll find why seeking out the softest games will dramatically increase your profits.
Simple Coin-Flip Games Show How Expectation Works
Flipping a coin could not be further from poker – though it does illustrate the concept of expectation simply. This can show the difference between your long-term results and short-term variance. I’ll assume this coin is fair (landing on heads and tails exactly 50% of the time in the long run).
If you were to get together with a friend, who offered you $10 a flip with this coin there are two outcomes. 50% of the time you would lose your $10, 50% of the time you would win $10. Simple so far. While on a single flip you would either win or lose, over 1000’s of flips, your results would be very close to break even. This illustrates that while your expectation (long term profit or loss) is zero, in the short term (say, 3 flips) your outcomes vary a lot. You might win $30, loose $30 or fall somewhere in between.
Now that same friend offers you $10 if you win, and you’ll only pay her $5 if you lose. This is a great deal. You can calculate your expectation as follows:
- 50% of the time you win $10
- 50% of the time you lose $5
Over 1000 flips, you win 500 times (+$500) and lose 500 times (-$250). Your net gain is $250, which is $2.50 a hand. If were offered this deal, each time that coin flipped your expectation would be $2.50 in profit – regardless of the result of any individual spin.
Simple Expected Value In Poker Calculations
There are coin-flip type situations at the poker table. For example, Ace-King is as good as a flip against pocket queens all-in pre-flop. The concept of expected value in poker comes to life in the common example of an over-pair against a flush draw.
You hold a pair of aces, and your opponent has a flush draw in this hand. When the chips go in on the flop, you have a 65% (approx.) chance of winning, and your opponent will suck-out 35% of the time. We also need to take into account money already in the pot to work out the expected value of each player. Here we will assume $100 in the pot, you bet $200 more, and get called.
Here are how the numbers look:
- 65% of the time, you win the $500 chip pot
- 35% of the time, your opponent wins
If you ran this same situation 100 times, you would win an average of $325 and your opponent an average of $175. You are making a positive expectation bet, every time you put $200 into the pot, you win $125 more. Your opponent is losing $25 every time they make the same play. Note that the dead money already in the pot makes a big difference here.
In the moment, you might feel like you have taken a major ‘bad beat’. Note that the negative expectation for your opponent is -8% (using approximate numbers). While this is a mistake on their part (and those 8% edges are important), it is not as terrible as you might think.
These are simple examples to illustrate the concept of expected value in poker. As you take into account multiple players, money left behind to bet (for example if the turn does not come up in the same suit as the drawing player), and the chance of the aces making a full house by the river even when the flush does appear – you’ll see the real-life calculations get complex.
Practical Applications of Expected Value in Poker: Starting Hands
Over a large sample of hands, making more +ev decisions than your opponents are exactly where the profit comes from in poker.
When you consider what starting hands to play, long-term expectation comes into play. You might already know that good players open with fewer hands from early position, compared to when last to act. The reason is that small pairs and lower Ace-x hands have a negative expected value in poker from early position. There are simply too many opponents with unknown hands to make them profitable.
Of course, expectation is only a guideline. If you are playing in a game where opponents play ‘face up’ (are easy to read) then you might have a greater expectation with a wider range than in a game with experienced players.
Long Term Profits and Expectation
You can use expectation to decide how to play individual hands profitably. This same concept can also be used to assess your long-term profitability.
If you know that you have an edge against players in your game, you can work out an hourly profit rate, based on how much you make in each hand over the long-run. We know that poker has a huge chance element. Even the worst players will sometimes win big pots, hitting a lucky card when they had a very slim chance of winning. Over time, superior (+ev) decision making will prevail.
If you make an average of 4x the big blind per table per hour – and play five tables, you know that you’ll make $20 an hour over the long run in a 50c / $1 game. If you suffer a bad beat and lose $200 this time, you might end the session down. This will balance out over time – the more +ev plays you are making, and the larger the number of hands in the sample, the closer to your ‘true’ expected value in poker you will be.
Bankroll Management and Expectation
Those chance elements in poker are the reason that good players manage their bankrolls conservatively. In order to remove the effects of variance from the game, you need to play a lot of hands. Having a big enough bankroll to ride out those swings is a prerequisite to being profitable in the long run.
Put another way, you will not have a chance to realise your real (positive) expected value in poker unless you have a big enough bankroll to ride out the short-term swings.
Cash game players ensure they have 20 buy-ins to make sure that suck-outs or ‘coolers’ do not dent their longer-term profitability. For tournaments the swings are much bigger, and 100x your average buy-in is recommended. Sit n go tournaments are in-between, with 50 buy-ins recommended.
Expectation Against Fish and Winning Players
The factor which has the single biggest effect on your expectation is how many mistakes your opponents are making. If you play against opponents who are constantly making negative expectation decisions, your profits will improve.
This is a dynamic factor. You might crush a $100 buy-in game – and be crushed yourself at a $500 buy-in table. It highlights the need to quantify your expectation against bad players, and to avoid ‘regulars’ (especially at online sites known for being easy) who may be taking money from the games.
Table selection is the key factor here. You should make sure you choose the games where your expectation is highest. If you see four regulars at a certain table, against whom you have only a tiny edge, then they will be taking money from the remaining bad players. By factoring in their edge against the players you are targeting, you will see that there is less profit to be made for you.
Even though you expect to have a positive expectation against the regulars – you should skip this table and find a better one. After all, if you have the fish to yourself, you don’t need to share the spoils of their negative ev plays with anyone else!