I'm at a roulette table. I only bet on red. When I lose I triple my bet, when I win I restart. Is this a roulette strategy?

This is the Martingale system, it will 100% work if you have infinite money and can bet infinite amounts.

So help me out here, what am I missing?

You're forgetting that not all outcomes are equal. You're just comparing the probability of winning vs the probability of losing. But when you lose you lose much bigger. If you calculate the expected outcome you will find that it is negative by design. Intuitively, that means that if you do this strategy, the one time you will lose will cost you more than the money you made all the other times where you won.

I'll give you a short example so that we can calculate the probabilities relatively easily. We make the following assumptions:

• You have $13, which means you can only make 3 bets: $1, $3, $9
• The roulette has a single 0. This is the best case scenario. So there are 37 numbers and only 18 of them are red This gives red a 18/37 to win. The zero is why the math always works out in the casino's favor
• You will play until you win once or until you lose all your money.

So how do we calculate the expected outcome? These outcomes are mutually exclusive, so if we can define the (expected gain * probability) of each one, we can sum them together. So let's see what the outcomes are:

• You win on the first bet. Gain: $1. Probability: 18/37.
• You win on the second bet. Gain: $2. Probability: 19/37 * 18/37 (lose once, then win once).
• You win on the third bet. Gain: $4. Probability: (19/37) ^ 2 * 18/37 (lose twice, then win once).
• You lose all three bets. Gain: -$13. Probability: (19/37) ^ 3 (lose three times).

So the expected outcome for you is:

$1 * (18/37) + 2 * (19/37 * 18/37) + ... = -$0.1328...

So you lose a bit more than $0.13 on average. Notice how the probabilities of winning $1 or $2 are much higher than the probability of losing $13, but the amount you lose is much bigger.

Others have mentioned betting limits as a reason you can't do this. That's wrong. There is no winning strategy. The casino always wins given enough bets. Betting limits just keep the short-term losses under control, making the business more predictable.

Another way of thinking about it is betting your entire bankroll for 99.9....% certainty that you will win $1.

Say you go into the casino with $1000.

Bet:

$1 lose.
$3 lose.
$9 lose.
$27 lose.
$81 lose.
$243 lose.
$729 oh wait you can't bet that much, you only have $457 left. Dang, do you bet $457 or find another $272? 

Bet $457 and you win $914! Congrats you're now only down $86!
Or maybe you lost and are down $1000.

Or maybe you scrounged up $272 so you could keep playing
Bet 729 and lose. Now you're down $1272.
Or
Bet 729 and you win 1458. Pay back the $272 you borrowed from your buddy, you're still up $186. 
You just bet $729 dollars for a %50 chance of winning $186.

But what are the chances of getting 6 or 7 losses in a row? 1 in 64, or 128 respectively, actually worse because roulette wheels aren't 50/50, they're 18/19 (18 wins and 19 losses in 37 plays on average) or worse. So losing 6 times in a row will happen 1 in 54 plays, 7 losses is 1 in 106.

Google says roulette wheels spin 55 times per hour so with your strategy you will lose your bank roll in about one hour assuming your starting bet is 0.1% of your bank roll.

I didn't get through all the comments, but most of the top ones and haven't yet seen the most basic casino check to stop these strategies is a table maximum. Same with doubling your bet each hand you lose at black jack, one bad run and you hit table max and you can't raise your bet far enough

To make things worse, Vegas now have triple 0 roulette tables. Three zeros in a row. In the same place. Do not play Roulette if you want to win. The only bet in a casino isn't to the casinos favor is on Craps and it's called ODDS. But to get to able to play it you have to win against 1.41% house edge first. And it is capped about 10x or 5x of original bet.

If you want to have the best odds against a casino that is hard work and not solid strategy. Counting cards can make money but honestly the amount of work it requires - you are better on the stock market or even in crypto.

Do not gamble against the casino.

The only reason to gambe is to have fun and safe risk with money you decided to spend and to never see again.

I start with 1 dollar
Game 1 - Bet $1 Lose

You are done at this point, having lost all your money.

It's the Martingale Strategy, except you're speed running the losses by tripling instead of doubling. I invented the Martingale Strategy on my own in my early twenties and thought I had figured out how to beat the house. I was pretty disappointed when I learned that 1. It already existed. 2. Would lose in the long run.

Tables have a bet limit. But there is no limit to how much you can lose.

The fundamental theorem of gambling is a mathematically proven fact which states that as long as the house has an edge, there is no betting strategy that has a positive long-term expected value.

This is what is known in the betting industry as "chasing your losses" and is a warning sign of a problem gambler who requires an intervention.

Congratulations, you've invented the Martingale betting system and are well on your way to becoming an adept probability theorist

The short answer is that casinos account for this by changing the profit returns and the odds of making a profit at all so that catastrophic losses are much more likely

With the first rise of online casinos (pre 2010), a friend of mine thought he had rigged the system by more or less the same strategy. I think the usual approach is actually to double the bet.

Well..in the end he won a debt of several 10k€ before he had to quit. This put a huge strain on his early adulthood.

Martingale

Run your simulation to find The expected number of bets before you go bankrupt. That will be illuminating

This isn't exactly the same as the Martingale system (where you double the bet after a loss so that you end up winning the starting bet when your colour does finally come up) but it's the same principle, just messier because you're tripling so it's harder to analyse (but you'll go broke quicker).

In principle, you will usually end up making a small profit while risking very big losses. And that's why it doesn't work. Either you run out of money for the next bet or you reach the house limit (which will almost always exist).

If you go to a casino with an amount of money you are prepared to lose, it's not the worst strategy (given that there are no good strategies with roulette). You will usually get a small win but you will sometimes get a big loss. You'll end up down in the long run.

Assuming you don't have infinite money, and that there are limits at the table, this is a good example of a non-ergotic system, where the mean and the median diverge. Because the rules of the game you defined dictate that at some point, you run out of money and have to stop playing, the outcomes for the median dip below that of the lucky few, who win more than lose owing to random chance. This is a super cool concept in a lot of systems, not just gambling fwiw

Check out https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3 or https://taylorpearson.me/ergodicity/

From what I've heard, this is why American roulette tables add a double-zero. Without that, the probability does slightly tip in favour of this system.