Expected Value of a Bet in casino games

Mar 5 2024
4 min. read
roulette 30
Ioannis Kavouras
By Ioannis Kavouras

What is Expected Value for casino games

Expected Value is the long-run average value of repetitions of the experiment it represents. This we usually call simply as “Expectation” and doesn’t tell us if and how much we can win or lose by a specific bet in a gambling game.

In gambling, by Expected Value or Average Expectation, we mean the calculation of how much money we will win or lose in the long run based on the odds and payouts of the game. Some clarifications are needed:

  1. The amount of money we should expect to win or lose is usually given as a percentage of the money wagered.
  2. By long run we mean after many thousands of trials. The more trials the closer we will get to the expected value prediction.
  3. This is a relatively accurate theoretical prediction of the long run result of our play, but still, actual results can vary.

The House Edge of casino games

Naturally, all casino games have a negative expected value. All players are expected to lose in the long run. The games are designed in favor of the casino. Actually, the expected value of any casino games is equal (in a negative way) to the house edge – they are the same thing. Specifically here is the expected value or house edge of the main casino games:

  • Roulette (double zero): -5,26%
    Roulette (single zero): -2,7%
    Roulette (even chance bets, single zero with La Partage or En Prison rule): -1,35%
  • Blackjack: -0,5%
  • Caribbean Stud Poker: -5,22%
  • Craps Don’t Pass: -1,36%
    Craps Come: -1,41%
    Craps Tie: -14,35%
    Craps Pair: -10,36%
  • Baccarat Banker: -1,05%
    Baccarat Player: -1,2%
  • Video Poker: -1% (varies – depends on the game)
  • Slot Machines: -10% (varies – depends on the game)

This negative expected value or house edge is the one and only reason that mathematicians claim that it is impossible to win a game of chance in the long run. I have explained with arguments my disagreement to this thesis, in the house edge monograph.

How to calculate the expected value of any game and any bet

Sometimes there can be special rules or maybe the is no fixed house edge. You can always calculate the probability and the expected value (negative house edge) of any bet by using the following equation:

expected value

If we have multiple bets at the same time: We multiply the probability of our bets to win with their respective profits in case of win and we add them together. Then we deduct the probability to lose multiplied by the loss (which is our bet amount). Then we divide the sum to the amount of our bet.

If we only make one bet, then the equation is:

expected value simple betNow to make things even more simple we can use as bet amount just 1 unit. Whether we bet more or less the expected value will be a fraction of our bet, so it makes things more simple to calculate if we use 1 (unit/chip/dollar/euro whatever) as our bet. In this case the equation becomes:

expected value math

Dozen Example

Now let’s calculate the expected value of the dozen bet in roulette. We already know (see previous section) that every bet in single zero roulette has an expected value (or house edge) equal to -2,7%, but let’s calculate it ourselves.

For a dozen, the probability to win is 12/37 because we are effectively betting on 12 numbers from 37 total numbers. The profit in case of win is 2 units. The probability to lose is 25/37, because 25 numbers out of 37 make us lose. So the Expected value of the dozen bet is:

expected value dozen

Which gives:

expected value dozen

Which means that “in the long run” we expect to lose an amount equal to 2,7% of the total amount wagered.

Fantasy Example

A question from our forum:

Let’s say the that I was playing the following numbers: 1-2-3-4-5-6-7-8-9-10 and was betting 1 chip on each number. Now the croupier congratulates me for my birthday and says “you can play two extra numbers where you don’t have to pay” and I add
11-12. So basically now I am playing 12 straight numbers for the price of 10. What would be my advantage?

Let’s calculate the expected value of this fantasy bet. We are betting 12 numbers with only 10 chips and each number pays 35 units profit. We have:

example

Which gives:

example value

So instead of a -2,7% disadvantage, in this special example we have a 16,7% advantage.

With the equation presented in this article you can calculate the advantage or disadvantage of any random or lucky bet, if you know the probability of winning.

About the Author See all Ioannis Kavouras Articles
Ioannis Kavouras
Ioannis Kavouras

I am a roulette player. Neither an “editor” nor a “mathematician”. The difference is that I put my money where my mouth is. Most roulette sites are written by editors who have never placed a bet. They recycle knowledge and recite Wikipedia. And they are paid to write. I have paid dearly for every single word I write. I have invested money, time, aspirations and grey matter in roulette.

In my 20 years as a roulette player, I have played everywhere, I have tried innumerable of roulette strategies, studied, created or dismissed countless ideas and done all the mistakes you could possibly do. I learned and I’m here to tell you about it. I can’t fight luck and i can’t control randomness. But I do know what works and what doesn’t, what mistakes you should avoid and what you need to win.

Roulette 30
In June 2010, disappointed by misinformation, outrageous claims, cheating roulette games and “sure win” system and device sellers, I created “Roulette 30”. I wanted to share knowledge, inspire other roulette players to make the next step in their game. It is as a very personal project. Everything, from the articles and the design to the last line of code is handcrafted with love by myself! I started it out of love for the game and since then Roulette 30 became a passion into itself.

At Roulette 30 we strive to create the deepest, interesting and inspirational site about the game of roulette. In this site I've shared as much as I can. Freely. Learn the basics of roulette, study the math and the physics of the wheel and then think creatively and discover new answers to old, classic roulette questions.

Roulette 30 Forum
This is the best place for serious roulette discussion. With a knowledgeable and friendly roulette community. I have already written almost 2K posts in the forum. Everyone is welcome to ask or comment on anything about roulette.

Kavouras Bet
After many years of people asking me about my personal roulette strategy, with reluctance and great care I decided to share all the details of the Kavouras Bet system. In that site you will learn everything you need to know about it.

These are the articles I have written I’m most proud of:

  • Kavouras Bet strategy
  • Roulette Wheel secrets
  • The Romanosky bets strategy
  • Bets and payouts of roulette
  • Analysis of Labouchere progression
  • Debunking the myth that the house edge is a game-stopper

Beyond roulette

I’m a restless soul, a novelty seeker, a different viewer.
I’m a Sagittarius with a touch of Pisces and Scorpio.
I was born in 1970.
I have studied Physics in the university of Athens, but never worked as a physicist.
I worked 15 years as a copywriter and creative director in advertising companies
I love Alfa Romeos, Italo Disco, ASIA, Nietzsche and dark haired women, though I have better luck with light haired ones.
I’m a loner and sentimental. I love philosophizing and reading. I’m smart, but I’m lazy, dilatory and a perfectionist, so I rarely accomplish anything.
That’s my story.
I would love to hear your story or thoughts at 30@roulette30.com.
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