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EV Calculations Tutorial 3: More Complicated Scenarios

If you haven’t read part 1 and part 2, then you’ll need to go ahead and do that before you jump into this week’s article because I’m going to assume that you know and completely understand the contents of those two parts.

This week, we’re going to look at some more complicated scenarios and show you how to expand your current knowledge of EV calculations so that you can calculate the EV of just about anything as long as you know enough information.

A Semi-Bluffing Scenario

Here’s a simple semi-bluffing scenario. Suppose we’re heads-up on the turn with a $40 stack and a $50 pot. Our opponent has us covered. If we go all-in, our opponent has a 25 percent chance of folding. If he calls, then we’ll have 33 percent equity. What is the EV if we open shove?

First, we have to determine all of the possible outcomes:

  1. Our opponent folds.
  2. Our opponent calls and we win at showdown.
  3. Our opponent calls and we lose at showdown.

This gives us the following equation:

EV of shoving = EV of outcome 1 + EV of outcome 2 + EV of outcome 3

As usual, we’ll need to find the EV of all three outcomes. The EV of outcome 1 is (0.25)(50) = $12.50 using the same process that we looked at in last week’s article. In today’s post, we’re going to be focusing more on the second and third outcomes.

Compound Probabilities

Let’s look at outcome 2. We need to know the chance of this outcome happening and the profit when it does happen. Our profit will be $90 since we’re winning the $50 pot plus the $40 that our opponent would call on the turn and lose to us. Finding the chance of this outcome happening, however, is a little more complicated.

For outcome 2 to happen, there are two different events that have to occur. First, our opponent has to call (75 percent chance). Second, we have to win at showdown (33 percent chance). These are two different probabilities, and we find the chance of both of them happening by multiplying them together. This means that the chance of outcome 2 happening is (0.75)(0.33) = 0.2475.

Since the profit of outcome 2 is $90 and the chance of it happening is 0.2475 (or 24.75 percent), then we find the EV of the second outcome by multiplying those values together to get $90 * 0.2475 = $22.28.

Now let’s find the EV of the third outcome. We know that to do that, we need to know the chance of it happening and the profit when it does happen. The profit is -$40 since we just lose $40 when we shove, get called and lose. So what’s the chance of it happening? We need two things to happen for this outcome to come about.

First, our opponent has to call (75 percent chance). Next, we have to lose at showdown (67 percent chance). To find the chance of both of those things happening, we’ll multiply the two chances together to get (0.75)(0.67) = 0.5025 which is the same as 50.25 percent.

Now that we have the profit of the third outcome (-$40) and the chance of it happening (50.25 percent), we multiply them together to get the EV of the third outcome, and that comes to (-$40)(0.5025) = -$20.10.

Putting It All Together

Like we have done before, we will find the EV of our semi-bluff shove by adding up the EVs of each of the possible outcomes:

EV of shoving = EV of outcome 1 + EV of outcome 2 + EV of outcome 3
EV of shoving = $12.50 + $22.28 + (-$20.10) = $14.28

So on average, it looks like we’ll have a very profitable semi-bluff shove in this situation.

Check Yourself Before You Wreck Yourself

Once you start dealing with more than two possible outcomes, you’ll have to deal with compound probability situation where you’re multiplying together the chances of different things happening. In the second outcome, we multiplied together the chance of Villain calling and the chance of us winning. Int he third outcome, we multiplied the chance of Villain calling with the chance of us losing. Since it can be a little tricky to figure out if we did everything right since there are a lot of numbers being pushed around, it’s a good idea to have a way to check to make sure we got the percentages right.

Here’s how to do it. Add together just the chances of each of the three outcomes. If you get 100 percent total, then you did that part right. If you don’t get 100 percent total, then you made a mistake. Here’s an example from above:

Total Chance = Chance of Outcome 1 + Chance of Outcome 2 + Chance of Outcome 3
Total Chance = 25 percent + 24.75 percent + 50.25 percent
Total Chance = 100 percent

Practice Makes Perfect

Dealing with compound probabilities is a little tricky if you haven’t done it a lot before. Because we’re going to be using it a lot in basically every single EV calculation that we do for the rest of these tutorials, you’re going to need to be able to do it. To this end, I want to give you guys some practice. If you look in the Beginner’s Circle forum, there will be a thread for this week’s strategy post. Post your answers to this homework problem in that thread. After a week or so, I’ll post a detailed answer of this problem so that you can check to see how you did and figure out any problems you need to correct.

Homework Problem

We’re heads-up with a single opponent on the flop. We have a $7.50 stack, our opponent has us covered, and the pot is $5.20. If we go all-in, our opponent will fold 38 percent of the time. If our opponent calls, we’ll win 41 percent of the time. What is the EV of our shove?

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