This week’s article is the beginning of a new series about expected value calculations. A lot of people have expressed an interest in having a good set of tutorials explaining how to do more math-related things since that’s something I’m kind of known for explaining here at FTR, so that’s what we’ll do for the next few weeks.
The Need for EV Calculations and What You Need to Know
First off, let’s talk about why these calculations are needed and why they can be useful. If you aren’t sure which play is the best between two options, then you can create a model for each of those plays and compare how well each model performs. This can give you a lot of insight about what the correct plays are like and what strong strategies look like even if you can’t create perfect models that reflect every nuance of the hand in question.
To be able to follow through this series, you will need to know a small amount of algebra. Knowing how to solve linear equations is enough to get you through 99 percent of basic EV calculations like the ones that we are going to be looking at here. Although not required, knowing how to use formulas to make basic spreadsheets will be pretty useful as well. This can help save you from a lot of repetitive math, and it can also make your study time more efficient.
How to Find the EV of an Outcome
To find the expected value of a situation, you add up the expected values of each possible outcome from that situation. The expected value of an outcome is the chance that it happens multiplied by your profit when it does happen. Let’s look at a quick example.
Suppose we’re rolling a fair, six-sided die. If it comes up as a six, we win $10. If it comes up as any other value, we lose $1.50. What are the expected values of each of the outcomes? Here I’m going to break down a simple process for setting up an EV equation that can be used on any type of situation no matter how complicated it may be.
First, we need to list out what the outcomes are. Here it’s pretty simple. The first possible outcome is that we win by rolling a six, and the only other outcome is that we roll a different number and lose. So far, our EV equation is just the sum of the EVs of those two outcomes. It looks like this:
Total EV = (EV of rolling a six) + (EV of rolling something else)
Second, we need to find the chance of each outcome happening and the profit from each outcome. For the first possible outcome, there’s a 1/6 chance of it happening, and our profit is $10. We multiply those two values together to get the EV of the first outcome. Here’s what our EV equation looks like at this point:
Total EV = (1/6 * $10) + (EV of rolling something else)
Now we have to find the chance of the second outcome happening and the profit that we have when it happens. There are five ways out of six for this outcome to happen, so the chance of it happening is 5/6. Because we lose $1.50 whenever this outcome happens, our profit is -$1.50. If we add those to
our EV equation, we get the following:
Total EV = (1/6 * $10) + (5/6 * -$1.50)
Now we have a basic math equation that we can evaluate to find the total EV of playing this game with the die. Here we go:
Total EV = (1/6 * $10) + (5/6 * -$1.50)
Total EV = $1.67 + (-$1.25)
Total EV = $0.42
So our total EV of rolling this die once like described above is $0.42. On average, we’ll earn $0.42 every single time that we roll.
Lessons to Learn From This Example
This example with the die may seem like an over-simplified game, but a whole lot of important situations in poker operate on the exact same principles with the exact same level of difficulty as far as the math goes. What we’re trying to do with this example is to get familiar with the process of finding EV equations. Once you’re familiar with the process, you’ll be able to set up equations for even more complicated scenarios in just a few moments. Consider the following steps:
- Identify each of the possible outcomes and set up an equation adding the EVs of each of them all together.
- Find the EV of each individual outcome by multiplying the chance of it happening times the profit you have when it does happen.
- Do the needed calculation with the numbers you found in step 2.
This is really all it comes down to. In fact, this type of organization in the process of finding the EV of complicated scenarios is often more important than being able to do some sort of complicated math problem.
Three Key Ideas to Know
Here are the three key ideas that you should know and be familiar with going into next week’s article.
First off, you should know that you get the expected value of a situation by adding up the individual expected values of each of the possible outcomes of that situation.
Second, the expected value of an individual outcome is the chance that it happens multiplied by your profit when it happens.
Finally, you should be able to organize how you find the expected value of a situation by using the three steps in the section above.
Overall, that’s really all you need to know to be able to know as a starting point. Next week, we’re going to use this process to find the expected value of some basic poker situations and use that as a foundation for getting into more complicated scenarios.