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.