A friend posed an interesting question today. He claimed a fool-proof gambling strategy for a game like Roulette, a game with just-shy-of-50% odds of doubling your wager.
His strategy was as follows:
- Start with an initial bet of $10.
- If you win, you’re up $10.
- If you lose, double the bet and repeat.
Assuming infinite money, a gambler can’t lose by using this strategy. Every losing streak is eventually broken by a win which compensates for the losses. For example, a 4-loss streak: −$10, −$20, −$40, and −$80, followed by a win of $160, exceeding the total $150 loss.
As it turns out, my friend had reinvented a betting strategy called a Martingale, which originated in 18th century France.
Vegas, baby!
Before you run to Vegas, though, there are some caveats.
To start off, this betting strategy requires arbitrarily large bets in order to overcome arbitrarily long losing streaks. That means you need:
- An infinite supply of money to bet.
- The ability to bet infinitely high amounts.
The first is mathematically difficult, and the second is hampered by casinos, which all have maximum bets on games like Roulette.
So what? Let’s explore.
Shifting Sands
The typical Roulette table has 47% odds of winning, a minimum bet of $10, and a maximum bet of $400. Let’s also assume you start with the modest sum of $1000.
Take a look at your holdings over time with this strategy:
Doesn’t look great.
I’ll be on a great winning streak when BAM suddenly I lose $630. Sometimes I come out ahead, but usually I lose before my 1000 spins are up.
The problem is that the maximum bet of $400 means that if I lose 6 spins in a row, I can’t compensate anymore by betting double — I’ve already bet $320 on that 6th spin.
“But,” the gambler in me complains, “losing 6 spins in a row is very unlikely! The chances are only 2.2%!”
“Yes, but,” the statistician in me replies, “you’re spinning 1000 times.”
The individual probability of 6 consecutive losses is only 2.2%. But I lose so much in that case — $630 — that I don’t make it up in expected winnings.
In a given round, where I play until I win or can’t double my bet, the probability of winning $10 is 97.8%; the probability of losing $630 is 2.2%. That makes my expected winnings per round: $10 × 97.8% − $630 × 2.2% = −$4.08.
Damn.
In 1000 spins, the probability of losing $630 at least once is 99.9992783% — and it only takes two such losses to bankrupt me.
To the Limit!
But even if the casino increases the maximum bet size, I’m only postponing the inevitable. True, I’ll go longer between catastrophic losses, but those losses will be much more catastrophic.
If you ask most people to come up with a random string of 1000 Ws and Ls, very few of them will include a 6-long streak. It just doesn’t seem “random” enough. But a truly random string will contain several such streaks, and longer ones too.
That’s as close to real randomness as your computer can get.
So what’s the lesson here?
True randomness is messier than you might think: events that are unlikely to occur in a single instance can become nearly inevitable in repeated iterations.