“Of course I don’t think there’s anything wrong with parties, or wanting to be married,” she pressed on, rather giving the impression she was arguing with someone in her head. “I just think there ought to be space for mathematics as well.”
“I’ll take your word for it. I’m afraid my own studies of the subject were limited by my capacity. Don’t despise me.”
“I don’t despise anyone but—well, really? Surely you are a mathematician?”
“Er...no?”
“But Uncle Hart said you play at the gaming tables.”
“Occasionally, as do most men,” Robin said, poised for defence or denial.
“Yes, but isn’t that mathematics in action? Or do you play dice, or roulette?”
“Cards. I prefer a contest of skill.” More to the point, he couldn’t cheat at the others.
“Good. Roulette is absurd. Are you familiar with d’Alembert’s system?”
“Who?”
“A French mathematician. He has a betting system for roulette which is based on mathematical principles—but I’m sorry, I’m running on.”
“No, no, I do want to hear it, very much,” Robin assured her. “Does it work?”
“Well,” Alice said, and was off in explanation. It sounded, as far as he could gather, like a martingale system for betting, in which one increased stakes after losing and decreased them after winning, though with some refinements.
“I’ve seen people play martingale,” he said. “Most of them swear by it, but quite a few of them lose.”
“Of course they do! Over the long run it might even out, but to follow his system properly would require infinite funds, which most people do not have—”
“I certainly don’t.”
“And it makes no sense at all in terms of probability, because the fall of the ball, or the dice, is not affected by what came before.”
Robin blinked. “You might have to explain that.”
“Say you roll a die six times, and the first five times it comes up one. Do you think it’s more likely to come up a six on the next throw?”
“It’s probably Greeked. I’d ask for new dice.”
She gave him a look. “All right, say it has been rolled ten times and everything except a six has come up. Would you wager on it being a six on the next roll?”
Robin thought about it. “Ten times? Probably. A six is due.”
“But it isn’t due. The die has no memory of what was rolled before, any more than a pack of cards remembers the last deal. Perhaps a six hasn’t been rolled in ten tries or even a hundred, but the odds of a six remain what they always were: one in six.”
“After a hundred?”
“Or a thousand. I grant you, if a die didn’t roll a six in a thousand rolls it might be—what did you say?”
“Greeked.”
“Greeked,” she repeated, with relish. “Assuming a fair die, though, it’s just a long string of chances.”
“But surely—”
They walked and talked for the next hour, Alice as animated as he’d ever seen her, eyes bright, not resting until she had persuaded him of d’Alembert’s fallacy, and then delving into the mathematics of other games of chance. Robin had good card sense and a long practical acquaintance with what he might expect in a game: he would not have expected an eighteen-year-old girl to dismiss his experience so confidently, but he found himself forced to listen as she chided him for superstition.
“But it doesn’t feel right,” he protested at one point. “Surely if the cards have been running badly for me, my luck—”
“Does not exist.” Alice had spoken strongly on the subject of luck.
“My chances will improve. A series of bad hands is bound to be followed by a good one.”
“Over time, of course. Over time, a fair die will come up on each side equally. But ‘over time’ is—oh, years, not an evening. Runs of cards, or dice, are inevitable, but a winning or losing streak is an illusion.”
“Then I’ve won a lot of money on things that don’t exist.”
“I expect you have. I’m not saying you can’t roll a six ten times in a row. I’m saying that when you do it’s simply a series of individual one in six chances.”
“But ten sixes isn’t a one in six chance, is it?” Robin said, attempting to work that in his head. “It can’t be.”
“No, of course not. It’s one-sixth multiplied by one-sixth multiplied by one-sixth and so on, ten times.”
“Wouldn’t that be very large?”
Alice gave him a kindly smile. “No, it would be very small. Fractions multiplied by other fractions become smaller. So it would be a very small chance indeed, but what you have to remember is that any specific outcome is just as unlikely. Suppose you rolled a die ten times and it came up three, one, four, one, five, nine, two, six, five, three. Would you see that as a streak?”
“How would I roll a nine?”
Alice waved a hand. “You’re using a ten-sided die. It’s an illustration. Answer the question.”
“I have never seen such a thing. And of course it isn’t a streak. It’s random numbers.”
“Actually, it’s pi.”
“It’s what?”
“Pi. Archimedes’ constant? The ratio of a circle’s circumference to its diameter? Surely you studied geometry at school.”
She looked genuinely puzzled, as if Robin ought to know who Archimedes was or why he was constant. “I didn’t always pay my tutor his due attention, I fear,” he lied smoothly. “What have circles to do with dice?”
“Nothing. All I meant to say is, as a matter of probability, it is exactly as likely, or as unlikely, that you should roll three, one, four, one, five, nine, two, six, five, three in that order as it is that you should roll ten sixes. Each combination is as unique and surprising and impossible to predict as the other.”
Robin attempted to grasp her point. “So Archimedes’ constant is as rare as ten sixes—”
“And so is any ten numbers in a specific order. Whatever combination of results you get is unique, and absurdly improbable once you consider all the other possibilities. It’s just that people don’t notice if they aren’t showy.”
Her voice was passionate, as though she truly cared about the injustice to a string of numbers. Robin turned to examine her face. She gave a little shrug.
“I’m going to be thinking about this for days,” he said. “I had no idea you were so clever.”
Alice blushed, this time seeming pleased rather than embarrassed. “Please don’t mention it to Mama. I’m not really supposed to talk about mathematics to—as part of my Season.”
To men, she probably meant. The devil with that. Robin had a sudden fantastical vision of them as husband and wife, she developing foolproof mathematics to give him the edge at the card table, he turning her dowry into hundreds of thousands. He cleared his throat, wondering if it was time to put his luck—his chances—to the test.
“Oh, there’s Florence waving at me,” Alice said. “Goodness, is it noon already? I had no idea we’d talked so long, I’m probably late. I’m so sorry for running on.”