Expected Value

Introduction to Expected Value

There are certain types of people who enjoy a whole lot of math, and then there are the majority of people who do not. For better or worse, a moderate amount of mathematical understanding can be very helpful as a player, but it’s usually never explained in a way that’s accessible to people who don’t like math or who aren’t familiar with it.

Instead, what we want to do here is provide a really straightforward way to understand a key idea from gambling math that doesn’t require you to actually understand much math at all.

Essentially, we’re looking at the math behind gambling for people who don’t really “do math” in some serious way. However, people who are more comfortable with math will still get a lot out of this.

In what follows, we’re going to break down the idea of expected value. This will include what it is, what it means, how to find it from a math perspective and what to do with it.

What is Expected Value?

There are a lot of terms from math that aren’t really clear in what they mean when you use them in the context of gambling. Expected value is one of those, but here is a definition that will work for virtually anyone:

• Expected value is the average amount that you will gain or lose on a wager or other type of event.
• This is usually expressed in terms of a currency like €5 for a gain of €5 or -€3 for a loss of €3.
• Listed as EV for short, expected value is never expressed as a percentage like what you see with payout rates, RTP or house advantage.

Calculating the expected value, or EV, of a bet or event is the single most basic and most fundamental piece of gambling math that you could ever do. What’s more is that it’s not very complicated at all once you get the hang of it, and it’ll allow you to perform your own analysis of atypical casino wagers, proposition bets with friends and more.

For a quick example of how EV could be expressed, imagine that you’re going to place a €10 wager on a slot with a 97 percent RTP. You’ll average losing €0.30 on each bet because the house edge is 3 percent, and 3 percent of €10 gives you €0.30. As a result, your expected value is -€0.30.

How to Find Expected Value

To find the total expected value of a wager or event, you need to add together the expected values of each possible outcome of that wager. That gives the following key concept:

• EV of Wager = EV of Outcome 1 + EV of Outcome 2 + …

From there, we need to know how to find the expected value of an individual outcome. We get this by multiplying together two numbers:

1. The profit we have from that outcome
2. The chance that outcome happens as a percentage

On a piece of paper or in a simple spreadsheet, we’ll need to know the profit of each possible outcome and the chance that each possible outcome has to find the total expected value of a wager. If we have these values in front of us, then it only takes a moment and a very, very small amount of really basic math (addition, subtraction and multiplication) to find the EV of that bet.

Expected Value Examples

We realize that this is a little tricky in the abstract, so we’re going to give a few examples here, some of which are more simple and some of which are based on real casino wagers.

In our first two examples, we’re going to use simple events while we break down exactly how you should think about this calculation. From there, we’ll jump right into it.

Example 1: Flipping a Weighted Coin

We’re going to flip a coin that’s been tampered with so that it lands on heads 60 percent of the time while landing on tails 40 percent of the time. You don’t know this, however, and your friend gets you to wager €10 per coin flip with you getting the worst of it.

The first thing we always do is make a list of each possible outcome. From there, we find the chance of each outcome happens and the profit for each of those outcomes.

In this case, we start with our two outcomes:

1. The coin lands heads
2. The coin lands tails

From there, we add the chance of each one happening and its respective profit:

1. Heads – 60 percent, -€10
2. Tails – 40 percent, €10

We multiply the chances of each one of these happening by their respective profits to get the EV of each individual outcome, which looks like this:

1. Heads – 60% * -€10 = -€6
2. Tails – 40% * €10= €4

Finally, we add these individual outcome EVs together to get the total expected value of the coin flip, which is -€6 + 4, and that comes to -€2.

What that means is that we will be losing an average of €2 each time we take this wager.

Example 2: Rolling a Six-sided Die

Suppose we have a fair six-sided die, and we’re going to win €5 if it comes up as a 1, win €3 if it comes up as a 2 and lose €2 if it comes up as any other number. We start by listing each possible outcome, its chance of happening and the profit for each.

1. Roll 1 – 1/6, €5
2. Roll 2 – 1/6, €3
3. Roll 3-6 – 4/6, -€2

Now, for each outcome, we multiply the chance of it happening by the profit to get the individual EVs for each individual outcome. We will be using a calculator for this, and we’ll round to the nearest cent.

1. Roll 1 – 1/6 * €5 = €0.83
2. Roll 2 – 1/6 * €3 = €0.50
3. Roll 3-6 – 4/6 * -€2 = -€1.33

If we add these up, we get €0.83 + €0.50 + -€1.33, which comes to €0. With this die roll, we will average breaking even.

While we know that this is a little bit of a simplified set of examples, what we want to show you is that the exact same process happens with more complicated wagers on real casino games.

Example 3: Betting the First Column in American Roulette

In American Roulette, there are a total of 38 spots on the board. You have the numbers 1-36 along with a green 0 and green 00.

We also have something that’s called a column bet that places a wager on 12 different values that are all in a row down the long way of the betting table. We’re going to use the process we outlined above to find the expected value of such a column bet.

First, we identify the two outcomes, which is pretty straightforward here:

1. The column bet wins
2. The column bet loses

The column bet pays at a rate of 2:1, which means if we wager €10, we will profit €20 if we win. In this scenario, we’re going to wager €10 and see what our expected value will look like accordingly.

From here, we need to know the profit of each outcome (which we listed above) and the chance of each one happening.

1. Column Bet Wins – 12/38, €20
2. Column Bet Loses – 26/38, -€10

Now we just multiply these values together to get the EV of each outcome, again rounding to the nearest cent:

1. Column Bet Wins – 12/38 * €20 = €6.32
2. Column Bet Loses – 26/38, -€10 = -€6.84

When we add together these final totals, we get €6.32 + -€6.84, and that comes to -€0.52. On average, we’ll lose about 52 cents for each one of these €10 column bet wagers on American Roulette.

Overview

Expected value is really straightforward to calculate when you use the format and approach that we have listed out above, and it has a ton of applications in strategy for casino games. With this, you can evaluate all kinds of different bets to figure out which will perform the best for you, even in games that do not have strategies or analysis published online.