Poker Equations
Expected value (EV) is a way of calculating how much we stand to make in a particular situation – it can be applied to poker or in lots of areas in real life.
Table Of Contents
Why Using a Poker Solver at Low Stakes is a Bad Idea Poker solvers are powerful tools to take your game to the next level. In fact most mid stakes and high stakes online poker pros these days have at least toyed with one at some point. Is poker a game of skill or chance? This question has been discussed and argued in many places and is the center of the arguments for and against legalizing Texas holdem and other forms of poker in many places, including online. The answer to this question boils down to the mathematics behind the game. EV = (probability of gain). (value of gain) + (probability of loss). (value of loss) EV = (0.5). ($42.5) + (1-0.5). (-$32) = $5.25. Therefore, betting $32 is going to be very profitable in this spot if our opponent folds 50% of the time – the reason for that: our opponent is folding too much. Poker math is NOT rocket science. The basics of calculating poker odds are actually quite simple and only require knowledge of addition, subtraction, multiplication, and division. If you made it past the 5th grade, you can learn to figure “pot odds” in no time. Personally, I played no limit Texas Holdem for YEARS without knowing ANY of this. Poker Mathematics. Poker is a game of skill and using the ability to read situations and opponents to give you the advantage in each hand you play. It is also a game of mathematics, where you should be able to calculate the odds of either you or your opponent winning the hand in any situation. The articles in the following table are the most.
Expected Value Definition
The definition of expected value is the average returns we would expect from taking a particular action (.i.e betting/raising/calling).
The expected value will be based on our current pot equity and pot odds (i.e. opponents bet size) when we face a bet and is based on our equity, our betsize and our oppoents fold frequency when we bet or raise.
If we are the player making the bet, the expected value will be based on our pot equity, bet size and fold equity.
Expected Value: Real Life Problem
To take a real world example – we park our car in a city and unfortunately it costs is $5 an hour. We know we will be at least 1 hour.
We estimate that the probability that we will be caught without a ticket is 10%. The fine for not having a parking ticket is $60.
Should we buy the ticket beforehand?
Or take the chance that we will not be caught?
To solve this problem we can use expected value.
We can find the expected value by using a simple equation. For people who aren't too keen on math, don't worry, it's pretty simple.
To work out our expected value we multiply the potential gain by the probability of that gain occurring (for example saving $5 by not paying the parking ticket by 90%).
Then we must find the product of the potential loss and the probability that the loss will occur and minus this value from the first part.
We have all this information from the real life example above:
Potential gain = $5, Probability of that gain = 100%-10% = 90%.
Potential loss = -$60, Probability of that loss = 10%
The wording may be quite confusing but the expected value formula will make more sense.
Expected Value Formula
Expected value, EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)
For our parking ticket example this becomes:
EV = (0.90)*($5) + (0.10)*(-$60) = $4.5 – $6 = -$1.5
So on average, every time we don't pay our parking ticket we will stand to lose $1.5.
This may confuse people as in no single case can we lose $1.5 – we either save $5 or we have to pay $60.
The $1.5 comes from is the average loss we will make over a long period of time. So if we did this 100 days in a row, 90 of the days we would have saved $5 each day for $450, but we would have been fined 10 times for $600 total.
Thus on average, we would have lost $150 which is $1.5 per day.
Change Variables and Find The Breakeven Expected Value Point
Using this simple calculation we can see how changing the variables affects our expected value; if it is less likely we will be caught and fined we should not pay for the ticket, that is obvious. For example, it is 1% likely that we will be caught without a ticket:
EV = (0.99)*($5) + (0.01)*(-$60) = $4.95 – $0.6 = $4.35. Therefore it makes sense to not buy a ticket for these parameters.
Intuitively that makes sense too since it is so unlikely we will have to pay $60 dollars.
If the fine cost less or the cost of parking cost more, that will increase the EV of not buying a ticket.
We can also find the point at which it becomes profitable to start not paying for tickets.
We find the breakeven point by setting the EV to zero and then find the probability that we will be caught which we have labelled as X.
This is called the breakeven point (requires some algebra):
EV = 0 = (x)*($5) + (1-x)*(-$60) => 5x-60 + 60x = 0
Therefore: 65x =60 and x = 60/65 = 92.3%.
So if we are likely to be caught greater 7.7% of the time we should buy a ticket; and if less than that we should not buy a ticket.
We can also apply this analysis to poker in situations where we know our equity pot odds, bet size and pot size.
Expected Value In Poker When Calling
To go back to the previous hand example with A9 of diamonds we had pot odds of 28% and we had pot equity of 18%. The pot odds are based on how much he bet and pot equity is based on the hands we assumed he would be bluffing with. So let’s put that all together to do an expected value calculation.
Again the expected value will be:
EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)
The probability of gain will be our equity which is 18%.
The probability of loss will be 1-equity which is 82%.
The value of our gain will be $67.5 and the value of our loss is -$26. Subbing in:
EV = (0.18)*(67.5) + (0.82)*(-$26) = 12.15 – 21.32 = -$9.17
Therefore this will be an unprofitable call to make. That should also be intuitive since we are winning so infrequently.
Expected Value When Betting
Here is another example of when betting on the river.
We opened to 3bb from UTG and got one call from the Button. On the flop of 9h7s3s, we have an inside straight draw and two overcards so we decide to continuation bet the flop and 3c turn.
Now on the river, we have to decide whether to bet again.
Firstly we can assume that we will lose if we check this river. We only have Jack high and unless our opponent has a busted draw AND checks, we will not take this pot down.
Therefore, checking has an EV of close to zero.
So what is the expected value of betting?
Well, we need to know:
- Our bet size = $32
- The pot size before our bet = $42.5
- How often our opponent calls = ?
How often our opponent calls is dependent on a lot of factors, but taking a rough estimate, let's say he will fold 50% of the time since we have shown a lot of strength and the river is a scare card.
We then the expected value equation becomes:
EV = (probability of gain)*(value of gain) + (probability of loss)*(value of loss)
EV = (0.5)*($42.5) + (1-0.5)*(-$32) = $5.25
Therefore, betting $32 is going to be very profitable in this spot if our opponent folds 50% of the time – the reason for that: our opponent is folding too much.
He should be calling at least 57% of the time to prevent us from making an immediate profit according to the minimum defence frequency formula which we will cover in the next section.
EV = (0.43)*($42.5) + (1-0.43)*(-$32) = ~$0
This J♦T♦ example is quite simple because we are on the river and there is only one more betting round to play.
However, it gets way more complicated when trying to estimate EV on the flop or turn for a number of reasons:
- You don't know what cards will come on the turn or river
- You don't know how your opponent will react to each of these cards
- There are so many different ways the hand can be played out, it's almost impossible to work out each path by hand.
For that reason, simple EV calcs like this are usually only performed on the river and software such as Cardrunners EV is used to simulate multiple street EV bets. Although these simulations take a lot of time to set up.
Minimum Defense Frequency
As mentioned, the minimum defence frequency is how often our opponent should call our bet to prevent us from making an immediate profit.
We can work our opponent's minimum defense frequency using this formula:
MDF = pot size / (pot size + bet size)
For the last example:
MDF = $42.5 / ($42.5 +$32) = 0.57 or 57%
Notice that increasing our bet size, decreases how often our opponent should call.
And the opposite is true too: decreasing our bet size increases how often our opponent should call.
A Quick Trick To Determine Profitability When Calling
A quick way to determine the profitability of a call without doing a full EV calculation is to compare the pot odds and our equity.
In the case of the example hand, we have pot odds of 28% and we have equity of 18%. If the equity is less than the pot odds we should not call as it would be unprofitable; conversely if we had more than 28% equity we can make the call as it would be profitable to do so.
You are not expected to do these calculations in their head while at the poker table. The purpose of the calculations is to analyse difficult hands off the table after your poker session, not during a game. This allows you to determine if you made the right decision and correct mistakes.
One of the main drawbacks of EV calculations is that you cannot perform them before the river as there are so many variables at play. We do not know what our opponent will do on future streets, will he check or will he bet? What card will come on the turn and river? If he does bet, what size will he use?
This is what makes poker such a complex game and a difficult one to solve computationally.
Despite this fact, expected value, pot odds and equity are useful in both poker and real life. So it is worth taking the time to understand how it is applied.
Expected Value Calculator
Make sure you check out this expected value calculator over at RedChip Poker:
It is extremely easy to use. All you have to type in are the three values in the above fields and it returns the expected value of that particular situation.
Here is a video which recaps the main points we covered in this lesson:
Closing Words
So that's it for our lesson on Expected Value. You should have already worked through the counting outs, pot equity and pot odds lessons before doing this so if not make sure you check them out.
You should now have a much better understanding of the math behind poker – your next step is to put it into practice and perfect it. Good luck!
Head back to poker 101 to learn more or check out the blog page for blog updates.
Poker solvers are all the rage these days. If you have never heard of a poker solver before, it is basically a computer program which helps you find the best GTO or 'game theory optimal' play in every situation.This allows you to balance your range in every situation and become an un-exploitable player. Sounds great in theory right? Unfortunately it doesn't actually work so well in reality at the micro stakes though.
In fact, I have already discussed before why using a GTO poker strategy at the micro stakes could actually even wind up hurting your results!
And in this article I am going to explain why using a poker solver might also wind up harming your results if you play at the lower stakes.
Why Using a Poker Solver at Low Stakes is a Bad Idea
Poker solvers are powerful tools to take your game to the next level. In fact most mid stakes and high stakes online poker pros these days have at least toyed with one at some point.
And many of them fully admit that they spend hours away from the tables every single day plugging ranges into a poker solver and analyzing the results.
And this is because in these games (NL200+), you are quite often playing against a lot of other very good players who are paying attention to every little move that you make.
Also, and this is extremely important, by the time you get to NL200+ online, the player pool starts shrinking tremendously.This means that unlike at the micros, you are probably going to be playing against the same handful of other guys day in and day out. And it doesn't even matter if you play on a really big site like PokerStars.
There just isn't a lot of action online these days at mid or high stakes. Instead, 90% of poker players are playing low limit games like 1c/2c, 2c/5c, 5c/10c and so on.
So basically what this means is that if you are in the 10% or so of mid or high stakes players, there are constant mind games going on because you see the same opponents again and again.
This means that you need to be constantly mixing up your lines and ranges in every situation to keep them always guessing.
At the micros though, as usual, it is a completely different story.
There are often thousands of players playing at the same stake on the bigger sites, and therefore you will rarely play pots against the same people.
And what's more, very few of them are even thinking on a high enough level for your mind games or tricks to ever work on them.
Exploitative Poker Is the Superior Strategy For the Micro Stakes
And this is because a GTO poker strategy simply isn't the optimal way to beat a bunch of low stakes poker players who are thinking on a very low level.
Most small stakes poker players are actually thinking on what we call 'level 1' as was discussed last week.
And basically what this means is that they are really only paying attention to the strength of their own hand. Many of them are not even capable yet of consistently thinking about what you might have.
Or even if they are, they aren't thinking in terms of ranges, they are instead still thinking in terms of individual hands. Which of course is the wrong way to approach the game.
So against most level 1 thinkers at the poker tables, you want to be looking for the best Exploitative strategy to beat them instead.
And what I mean by that is finding the perfect counter to every mistake that they make.
For example, maybe they value their over-pairs too much. This is something that is very, very common with low stakes players.
Many of them are simply incapable of folding a big pair like:
- AA
- KK
Against players like this, you want to see as many cheap flops as possible against them with speculative holdings like:
- 55
- 76 suited
And the reason why is because your implied odds are over the moon if you manage to flop two pair or a set.
Implied odds in poker by the way is just a fancy term that we use for what you stand to gain or profit if you hit the flop hard.
Versus a guy who doesn't fold his AA for any amount? Ya, our implied odds are basically infinite!
And so from there, you want to fast-play your big hands against the generally passive players at these limits in order to make sure that you stack them.
This is something that I discuss in massive detail in Crushing the Microstakes which is quite possibly the most 'Exploitative' and unbalanced poker book ever written.
And I am actually quite proud of that because these are also the same strategies that allowed me to win tens of thousands of dollars in these games.
There is no balance necessary at these stakes. There is no need to become un-exploitable either. And this renders a poker solver pretty useless at these limits.
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Using a Poker Solver Will Harm Your Results at the Micro Stakes
In fact, using a poker solver will probably end up having a negative impact on your results at the lower stakes.
Because for example, it is likely going to tell you to balance your flop play when you hit your set against the micro stakes guy who can't fold his AA for any amount.
The solver will recommend perhaps that you just call his continuation bet 50% of the time and raise the other 50% of the time.
Now, this is a great strategy against a world class level 3 or level 4 thinker at mid or high stakes.
But against a level 1 thinker at the micro stakes who is completely incapable of folding an over-pair, calling 50% of the time here is actually just throwing away EV (and ultimately money).
Instead, the correct Exploitative play in a situation like this is to literally raise your set 100% of the time. It doesn't matter how unbalanced your play is because your opponent isn't thinking on a high enough level to understand this.
Conversely, a poker solver will also advise you to bluff raise the flop sometimes in order to balance your range and become un-exploitable.
Again, this is a poor strategy that will not work against a calling station level 1 thinker at the lower stakes. They are just going to call you down and make you look silly.
You don't need to worry about playing un-exploitable poker at the micro stakes because most of the time your opponents are not paying attention.
And even if they are, unlike at mid or high stakes, there are hundreds or thousands of other players for them to keep track of as well.
So there is almost no chance that they are taking meticulous notes on your game like a high level pro will be at NL200+.
When Should You Consider Using a Poker Solver?
So when should you actually think about using a poker solver then? Well, I think it is probably already clear given everything I have said throughout this article.
You should start using a poker solver to improve your game by the time you play mid stakes online or live. I would put this at NL200 online which is the $1/$2 blinds game. Maybe around $10/$25 if you play live.
This is the first level online where a pretty good chunk of your opponents will be professionals and there are very few weak spots.
NL200 is also the level of online poker where the player pool starts to shrink drastically and you are going to be playing against the same handful of players day in and day out.
This is why it will be important for you to start balancing your play like the solver will advise, flatting your set 50% of the time, and raising 50% for example.
You also need to be bluff raising a solid percentage of the time as well so that you aren't just raising monsters like sets all the time.
This is all the kind of stuff that a solver will help you with. And it is absolutely crucial that you start learning how to play an un-exploitable style like this once you get to mid and high stakes.
There are tons of advanced poker training programs available these days to help you with this as well.
For example, the brand new Advanced Cash Game Strategy course taught by high stakes online poker legend Kanu7 is entirely based on solver theory and analysis.
Poker Equations Calculator
I had a chance to have a look at it already, and it is truly brilliant stuff. Alex 'Kanu7' Millar literally spent an entire year analyzing high stakes poker hands with a solver to create this course.But again, I would not recommend using these strategies unless you are playing against high level opponents in mid to high stakes games.
Because the exact opposite highly unbalanced Exploitative strategy is usually optimal if you play at the micro stakes. And especially if you play in the really low stakes games like NL2, NL5 or NL10.
Using a poker solver at these limits will almost certainly not help you. And what's worse, it might even harm your results.
Final Thoughts
Poker solvers are all the rage these days and for good reason. They are an excellent tool to take your game to a new level by helping you balance your range and become un-exploitable in all situations.
This a crucial aspect of your game if you want to have success against other high level thinking pros at mid and high stakes these days.
Poker Equations
However, the main problem though is that 90% of people don't play at these limits. They play against the level 1 thinkers at stakes like NL5 instead ($1/$2 live cash games or low buyin tournaments).The players who you are going to encounter in these games almost always only play part time as a hobby and many of them are still learning the basics of the game.
Most of them are not capable yet of thinking in terms of ranges and so trying to 'balance' anything or become un-exploitable will simply fly right over their heads.
And what's worse, the solver will actually have you making -EV (negative expected value) decisions in some spots such as slow-playing a set against a calling station.
Poker Equations Definition
Instead, an Exploitative unbalanced strategy is almost always the way to achieve the maximum success especially in extremely low limit micro stakes games like NL2, NL5 and NL10.