The Calculus Diaries - Jennifer Ouellette [33]
How do we determine that likelihood? We take an integral. The actual formula for the integral from one point to another on a bell curve has never been explicitly written down; it is usually calculated with a computer. But here is the gist of the concept. Imagine a number line that runs from negative infinity (on the left) to infinity (on the right), with 0 smack in the middle, and a standard bell curve peaking at 0. This represents the distribution of outcomes for a 50/50 chance of winning or losing. The probability of losing will be the area under the curve that spans from minus infinity to 0, while the probability of winning will be the area under the curve from 0 to infinity. Each is equal to one half in this simplified example. The more times we roll the dice, the closer we will come to matching those probabilities. With 50/50 odds, for an infinite number of rolls, we will break even.
We can be even more specific by picking a random point on the x axis—say, 500—to determine the likelihood that we will either lose money or win up to $500. The answer will be the area under that portion of the curve that runs from negative infinity to 500. If we want to know the likelihood that we will win more than $500, we determine the area under that portion of the curve that runs from 500 to infinity.
The biggest problem when it comes to craps is that the odds are not 50/50. Let’s say the house has a slight edge, making the odds 49/51. Now our bell curve is shifted slightly to the left on our grid, making it slightly more likely that we will lose; and the longer we play, the closer we will get to that distribution. We also need to specify the size and type of bet for each roll, because the probabilities in craps are linked not just to the outcomes of the rolls of the dice, but also to the payoff rates for different kinds of bets.
GAMING THE SYSTEM
We won at craps because we got lucky in the short term: We hit a probabilistic sweet spot by pure random chance and had the sense to quit while we were ahead. Vegas notoriously attracts gamblers convinced they have discovered a “system”—a perfect strategy to beat the house. They are deluding themselves. Even assuming these perennial optimists have taken every single variable into account for their calculations, it takes only the tiniest house advantage to tip the scales irrevocably. We played for just one hour. Play the game long enough, and eventually you will lose everything. The occasional perceived hot streak or lucky break doesn’t alter that fact. The casinos are very up front about this. Another craps dealer in the New York, New York casino—let’s call him Vito—didn’t mince any words on that score: “Everyone thinks they got a system. You think you’re gonna beat this table? Go ahead and try. We got ATMs all over the casino, just for people like you.” Listen to the wisdom of Vito, my friends. Forewarned is forearmed.
Even if the odds are in your favor, there’s no guarantee you’ll win. Let’s imagine the situation were reversed, and the players had the slightest advantage; it wouldn’t necessarily translate into an automatic win. You must pay just as much attention to your bankroll as the odds of winning; if the odds are good but you’re betting a substantial portion of your bankroll on each roll of the dice, it’s enough to wipe out any advantage pretty quickly. That’s the essence of a little exercise called gambler’s ruin. It’s a favorite of University of Washington physicist Dave Bacon—better known to