Pascal's roulette wheel, or how I won $20

I was at a casino last week. I was down on my luck, having lost $20 at the 7-card poker table.

Then I remembered when my friend Jesse had told me about the Martingale system which guarantees you to win at roulette! It goes like this:

First bet $20 on red. If red comes up, you’ve won. Quit
If you lose, double your bet to $40 and put it on red. If red comes up, you lost $20 the first round, but then won $40 the second round, so in total you won $20. Quit
If you lose again, double your bet to $80…
…and so on. Just keep betting red, doubling your bet each time, and eventually you are bound to win, taking home $20

This was clearly the only way to climb my way out of this $20 hole I was in. Given the table limit, the only way I could lose was if I lost roulette 5 games in a row (1). The odds of this were 4% - practically zero!! (2)

So I cashed in with $100 worth of chips and put $20 on red. I lost the first bet, but then bet again at $40 and won. I had won back at roulette the $20 I had lost at poker!

Jesse was a genius! I was half considering simply quitting my job and moving into a mattress in the corner of this casino to make my way as a roulette player.

Before making any final decision on my career and housing, I decided to crunch some numbers and calculate the expected value of carrying out the Martingale strategy into infinity.

As I should have expected given that casinos remain in business, the house still wins in the long run even if you use a Martingale strategy. This wasn’t a viable career option for me

The thing I hadn’t really considered was that if I lost 5 games in a row, I would have been losing around $600. Even with only a 4% chance of losing that much, this potential loss outweighs the 96% chance that I win $20 (3).

In my mind, 4% rounded down to “basically zero”. But in real life, things with 4% probability happen about 1 in every 25 times. In finance / probability, these are known as “tail risks”. And this tail risk had a disproportionally large consequence - losing more than $600.

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DALL-E’s interpretation of Blaise Pascal playing roulette with God in a Chinatown casino

Even if there was no table limit and I could play the game infinite times, I still would be expected to lose. This was super hard for me to intuitively understand. The odds that red loses 20 times in a row are 0.0002%. If I had enough money to double my bets 20 times, that feels like that’s basically a guaranteed win, right?

No. What feels like a “guaranteed” win is actually only a 99.9998% chance. This is a small tail risk but with a huge consequence - if I did lose 20 games in a row, I would be losing $20M (4). Which, despite my lucrative recent side hustles in forex and casinos, is more money than I have.

I was falling victim to failures in thinking about tail risks that Nassim Nicholas Taleb describes in The Black Swan: “just as we tend to underestimate the role of luck in life in general, we tend to overestimate it in games of chance.”

This is the first time I’ve been confronted with tail risks in a way that feels salient. I think having played roulette will help me when thinking about things like existential risks, risky but high-impact ventures, and the existence of aliens.

So now I have a good excuse to keep going to the casino. I’m not throwing my money away. I’m on an epistemic quest to sharpen my intuitions about low-probability events.

It also makes me think of Blaise Pascal, who I think would have happily taken my money as I played roulette over and over again, delighting in the small chance of winning huge amounts from me.

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Thanks to Binx for co-writing this post!

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1. the table had a maximum bet of $500, so if I lost at $20, $40, $80, $160, and $320, I would not be able to double my bet again to $640

2. it was a single-zero roulette table, so the odds of winning on red were 18/37. Odds of losing 5 in a row were (19/37)^5 = 3.6%

3. Expected value: $20 * 0.96 - $630 * .04 = expected loss of -$6

4. $20 * 2^19 = $10M lost on the last bet, then another $10M for all I lost cumulatively on the previous bets