Roulette: What To Expect?

Roulette is a staple of the casino floor and a favorite for beginner gamblers. It is simple and requires absolutely no skill. Unsurprisingly, America Roulette has a pretty solid house advantage. Over the long run, as a player, you will lose.

For a quick refresher on how the game works:

The dealer spins a wheel with 38* numbered spaces on it. 18 are red, 18 are black and 2 are green. They are numbered 0-36 and 00. The dealer tosses a ball into the wheel and it lands on a one of these spaces with a color and number.
You can make a variety of bets before the spin based on where you think the ball will land. You can bet on high/low numbers, odd/even numbers, a specific number, a set of 3 numbers, or a specific number
All bets have the same expected payout of -$0.053 per $1

*European roulette uses 37 spaces and has only 1 green space. They logically didn't need a 00. They also have better odds for the players.

What Should I Expect to Win

You shouldn't. End of post.

Just kidding. But you are going to loose on average. But that is not why people play. At first glance, its close to a 50/50 chance of doubling your money. Yes, the house wins more, but if you feel lucky you might walk away with more. After all, you don't actually lose 5 cents a game, you either win a dollar or lose it. It is the variance that makes it interesting.

So let's take a scenario where you start with $100 and make $20 bets (this is often the minimum on a weekend in Vegas). So after your first hand, you'll either have $120 or $80. You will continue to place $20 bets as long as you have money. After 5 rounds you will end up in one of these situations, with the following probabilities:

So after 5 rounds, you will have between $0 and $200. But you only have a 2.4% chance of doubling your money, but also only a 4% chance of being broke. You also have a 45% chance of having more money than what you started with.

How the Rest of the Night Will Go

I will revisit "average winnings" later, but lets focus on three probabilities:

The probability you bust and go home with no money
The probability you have more money then when you started
The probability you double your money

So after 30 rounds (maybe an hour? very casino dependent), you have a 46% chance of being bust, 30% chance of being up, but an 11% chance of having at least double your money.

The driving force of this graph is increased variability increasing with the first rounds and an overall average winnings that is decreasing over time. Around round 45 probability of being over $200 is the highest, and then the overall losing trend takes over.

There are a few additional quirks about this graph worth noting:

You can only go bust on odd rounds. This is because when you are on your last $20, you can either go bust or back up to $40, taking two more rounds to go out.
Probability of being up and over $200 bounce back and fourth around a steady trend. This is because we picked a steady threshold. Note that after the first round, you can't have $100, you either have $80 or $120.
Roulette is memoryless. This simulation is not. If after 100 rounds, you are back at $100 dollars, your odds are exactly the same as when you started. This simulation picks a starting point as a benchmark to compare winnings to. To understand this, let's go back to average winnings.

Back to Expected Value - Average Winnings

Here is the graph of expected value over that same time:

Notice that after the first five rounds, the slope of the line starts to level out. Why? Does this mean the longer you play the more you win? No. It does not. It means we are ignoring all the people who already lost and went home. When you have $0, you don't get to play again and have a chance of winning either $20 or going to -$20. You don't get to play.

If you did buy in again, your expected value would decrease at the same constant rate if you never had a hard stop.

Final Thoughts on Roulette

Pick an amount to lose, pick an amount you want to walk away with, and leave when you have no money or hit your threshold. Don't trick yourself into thinking you are up and can't go down or that you are on a hot streak. Gotta know when to fold 'em.