At some point, you’ve probably heard an urban legend like this: someone walks into a Las Vegas casino with his life savings, converts it into chips, and bets it all on one spin of the roulette wheel. In the version where he wins, he walks out very wealthy. But, in the more likely scenario where he loses, he leaves totally ruined. This isn’t just an urban legend, by the way, but it has actually happened numerous times— for example, in 2004, someone named Ashley Revell did this with $135,000, and ended up doubling his money in one spin at roulette. I most recently read about this incident in a book called “Chancing it: The Laws of Chance and How They Can Work for You”, by Robert Matthews. And Matthews makes the intriguing point that if you are desperate to significantly increase your wealth ASAP, and want to maximize the chance of this happening, what Revell did might not be entirely irrational.
Now, before we get into the details, I want to make it clear that I’m not recommending casinos or condoning gambling. Occasionally while on vacation my wife & I will visit a casino, and here’s my foolproof winning strategy. First you walk in, let yourself take in the dazzling atmosphere of the flashing lights, sounds, and excitement. Then walk over to the bar, plop down ten bucks or so, and buy yourself a tasty drink. Sit down at the counter, take out your smartphone, start up the Kindle app, and read a good book. (The Matthews book might be a nice choice, linked in the show notes at mathmutation.com.) Then relax in the comfortable knowledge that you’re ahead of the casino by one pina colada, which you probably value more than the ten dollars at that particular time. Don’t waste time attempting any of the actual casino games, which always have odds that fundamentally are designed to make you you lose your money.
Anyway, getting back to the Revell story, let’s think for a minute about those rigged odds in a casino. Basically, the expected value of your winnings, or the probability of winning each value times the amount of money, is always negative. So, for example, let’s look at betting red or black in roulette. This might seem like a low-risk bet, since there are two colors, and the payout is 1:1. When you look closely at the wheel, though, you’ll see that in addition to the 36 red or black numbers, there are 2 others, a green 0 and 00. Thus, if you bet on red, your chances of winning aren’t 18/36, but 18/38, or about 47.37%. That’s the sneaky way the casinos get their edge in this case. As a result, your expected winnings for each dollar you bet are around negative 5.26 cents. This means that if you play for a large number of games, you will probably suffer a net loss of a little over 5% of your money.
So let’s assume that Revell desperately needed to double his money in one day— perhaps his stockbroker had told him that if he didn’t have $200,000 by midnight, he would lose the chance to invest in the Math Mutation IPO, and he couldn’t bear the thought of missing out on such a cultural milestone. Would it make more sense for him to divide his money into small bets, say $1000, and play roulette 135 times, or gamble it all at once? Well, we know that betting it all at once gave about a 47% chance of doubling it— pretty good, almost 50-50 odds, even though the casino still has its slight edge. But if he had bet it slowly over 100+ games, then the chances would be very high that his overall net winnings would be close to the expected value— so he would expect to lose about 5% of his money, even if he put his winnings in a separate pocket rather than gambling them away. In other words, in order to quickly double his money in a casino, Revell’s single bold bet really was the most rational way to do it.
And this has been your math mutation for today.