The Calculus Diaries - Jennifer Ouellette [32]
Now it’s time to play the game for real. The casino graciously sets up a small-stakes table just for the newbies for one hour so we can practice our newfound skills. S-Money being the more math-savvy in our household, he shrewdly opts to take Dominic’s advice and maximize the size of his odds bets in relation to his line bet, thereby reducing the house edge to a whisker (although not eliminating it entirely). Because we’re experimenting, I choose to focus on placing come bets and point bets, trading better payoff odds for more control over the board, to see how these two strategies compare.
From a psychological standpoint, craps is an ingeniously designed game. The odds are certainly rigged in the casino’s favor, but they are not rigged too heavily in that direction. That would be no fun at all. Players need a sense of reward, even if it’s just the illusion of winning once in a while. That’s what makes the game so addictive. I soon notice that we do win rolls, sometimes several in a row, but in all but a few cases, the money we win never quite adds up to the money we spend placing even the minimum bets. The result: At best, playing craps is a slow bleed that one can easily not notice, because the tiny amounts won in between losses blind players to the long-term financial hemorrhage that is taking place.
Yet somehow we walk away one hour later with combined winnings of $145. (I win $45, S-Money wins $100—a neat illustration of how the various odds play out, albeit purely anecdotal.) My interpretation is that we hit a streak of good luck, and had the sense to quit playing while we were still ahead. Ever the physicist, S-Money loftily informs me that, according to probability theory, “hot” or “cold” streaks are merely a perception. Each roll is independent of the previous and subsequent rolls; that is the nature of true randomness. So the odds are the same for each roll, even if the shooter has rolled the point twenty or two hundred times in a row; the outcome of the last roll does not affect what happens next. There is no such thing as being “due” for a win (or a loss).
Still, it is possible to figure out how often we are likely to have a winning session of craps. Translating this concept into actual calculus is tricky, in part because throwing dice falls into the realm of discrete events—analyzing event probabilities under repeated trials—whereas calculus, by definition, deals with continuous things. If you plot the probability of the outcomes of individual rolls, you’ll get a shape resembling a pyramid—a perfectly good shape, but not one that represents a continuous function.
However, if you throw the dice two thousand times (or more), add up how much you win and how much you lose each time, and plot it all out on a Cartesian grid, the result is a standard bell curve, also known as a normal distribution curve. For any random sample—say, many random rolls of the dice—you will get a distribution of values clustered around an average (or “mean”) value. That mean value is the peak, or highest point, of the curve, where the most data points cluster together; there are fewer and fewer data points as we move out to the edges. In craps, big wins or big losses happen very infrequently, and would be found at the extreme edges of the bell curve, while outcomes with smaller wins and losses would cluster near the peak of the curve.
Now that we have a pretty bell curve, we can use calculus to determine how often we will have a winning session in craps. First we need to understand what it is we are calculating; we have to set up the story. Every time we throw the dice, the probabilities for rolling a specific number don’t change from the odds outlined above; there is still a 1 in 6 chance of rolling a 7 with each and every roll, as represented by the pyramid. We are asking a different question: Given that we know we will throw the dice two thousand times, what is the probability that we will win or lose?
This reduces the question to an either-or option, assuming even odds—and remember that the