Bonuses and Expected Value

The purpose of this page is to explain the relationship between expected value (EV) and casino deposit bonuses. There are plenty of gambling blogs out there that list the latest bonuses, but I’ve noticed a dearth in websites that actually explain the EV of bonuses.

It’s good to know the EV of gambling bonuses because the EV tells you if the bonus is a good deal or not. If you go around getting deposit bonuses without considering the EV, you’re leaving money on the table.

After reading this article, you’ll know exactly what I’m talking about when I list the EV next to any bonus description on the blog.

A Basic Explanation of Expected Value

Expected value is a concept in probability that describes the average outcome of a random event. In gambling games, expected value is calculated to determine the long-term profitability of any wager or scenario.

A positive expected value (+EV) means that the gambling proposition is profitable over the long term. To calculate the expected value of any gambling game, you need just two key pieces of information:

  1. The odds of each outcome occurring
  2. The payouts of each outcome

With that information, you can tell if any gambling game, wager or bonus is profitable or not. Even better, you can tell exactly how profitable that game is.

Coin Flip Example

Let’s use a simple coin-flip game as an example. Pretend I offer you a wager in which we flip a coin. I’ll pay you $1 if it lands on heads and you will pay me $1 if it lands on tails. You can probably already tell that this is an even-money game; you have neither an advantage nor disadvantage in this game.

But let’s dig into the math for a second anyways to illustrate how expected value is calculated and how to interpret the results.

There are two possible outcomes in this game and we know the odds of each outcome. We also know the payout of each outcome. Thus, we have all the information we need.

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What we do next is multiply each probability by each payout and then add it all up:

(1/2)*1 = .5


(1/2)*-1 = -.5

Total: 0

The expected value in this game is $0. That means over the long run, you can expect to break even in this game. Note that you will never win $0 in any single coin flip. You will always either win $1 or lose $1. Similarly, you will probably not come out exactly break even after 10 flips of the coin. In the short term, variance produces all kinds of crazy results.

But as we stretch this game out across many, many trials, your real results will reach closer and closer to exactly $0.

Don’t be turned off by how long this simple example looks in written form. I tend to be long-winded. The concept itself doesn’t require any advance mathematics.

Applying Expected Value to Casino Games

Here’s where things get interesting for gamblers. You can figure out the expected value of any casino game with a simple calculation. There are only two things that you need to know:

  1. The total amount of wagers that you plan to place
  2. The house advantage of the game

You can calculate the EV of any casino game by multiplying the amount of money that you plan to wager by the house advantage. For example, let’s say that you play a slot machine that has an exact house advantage of 5%. Let’s also say that you plan to spin the reels 1000 times for $1 per spin.

The total amount of wagers that you’re planning on wagering is $1,000. Multiply that by 5%:

$1,000 x 0.05 = $50

The expected value of this game is $50 in losses. Over the long run, you will lose $50 for every $1,000 that you wager in this particular slot machine.

I think it’s awesome how simple this calculation is. The house advantage for every casino game is published all over the internet. All you have to do is search for the house advantage of any game and multiply that number by the total sum of wagers.

In some games, the house advantage is affected by how you play. For example, the house advantage in blackjack changes drastically based on your strategy for that game. If you play a good blackjack game, you can get the house advantage down to 0.5% (and sometimes even lower). If you play blackjack poorly, the house advantage explodes upwards.

What this Means for Bonuses

What this all means is that you can figure out the expected value for any casino bonus if you have the following information:

  • The wagering requirements
  • House advantage of the game you will play
  • Size of the bonus

Whenever you have a casino bonus that you want to get, look for the wagering requirements. Figure out exactly how much money you will need to wager to clear the bonus. Then, look up the house advantage for whichever game you will be using to clear the bonus.

All you have to do next is multiply the house advantage of that game by the total wagering requirements.  For example, you plan on playing a 2.7% house advantage game and you need to wager a total of $60,000 on that game:

$60,000 x .027 = $1620

This means you can expect to lose $1620 over the course of clearing the bonus. You actual results may vary (by a lot), but this is the expected cost over the long term.

Now, compare that number to the amount of bonus money you stand to win. If this is a $2,000 bonus, you subtract $1620 in losses from the $2000 in bonus money for a grand total of $380.

The expected value of this bonus is $380.

In this example, the bonus is a good deal. Hooray! Not all bonuses are so generous, though. And that is why it is good to understand the relationship between bonuses and expected value.

Please see this blog post for a more detailed explanation for calculating the EV of bonuses.