Introducing a 7-part series: “The Bear in the Moonlight: Stories and Lessons in Probability”
Probability is the practice of quantifying uncertainty. It harnesses the power of mathematics to deal with our doubt, our ignorance, and our lack of guarantees in life. Like all of the best math, probability is brimming not just with practical applications, but with lovely ideas. It’s like a gorgeous painting that also functions as a dishwasher.
Probability is beautiful, useful—and oh yeah, totally befuddling to most people who confront it. Consider some of the obstacles:
1. Probability is often counterintuitive. When dealing with uncertainties, our minds struggle to overcome built-in biases that run counter to logic and reason.
2. Probability is not algorithmic. In algebra, just memorizing the right steps will take you a long way (towards getting right answers, at least). But in probability, each question is a world unto itself.
3. Probabilities are expressed as fractions, which vex plenty of people in their own right.
4. Probability demands comfort with very small and very large quantities. For example, suppose I’ve got 20 different novels in a box. If I remove them randomly, one by one, what’s the probability that they emerge in order from shortest to longest? Roughly 1 in 2 quintillion—a number beyond the typical limits of the human imagination.
5. Probability builds on combinatorics—the mathematics of sophisticated counting. Probability courses often begin with an intimidating unit on combinations, permutations, and the like. Conceptually, it’s the right starting point. But pedagogically, it’s awfully deep water for students just learning to swim.
Usually, someone learning probability tackles all these challenges at once. My hope is to isolate the first two obstacles: to help you wade into the non-algorithmic, counterintuitive nature of probability without getting drawn into the riptide of combinatorics and computations.
That’s where the stories come in.
Narrative engages the mind. I’ve seen it happen in the classroom. A little story—even a clumsy or tangential one—grabs students in a way that few lectures do. Whereas concepts are so smooth that they slip right through our fingers, stories give us texture, a rough surface to grasp. Once engaged, we find our intuition and critical faculties (too often dormant in math class) hum to life. We’re ready to wrestle with the big ideas, rather than crying “Uncle!” at the first sign of resistance.
Let’s be clear. You won’t learn probability just by reading stories. That’ll take teachers, puzzles, struggles, and most of all, time. But I humbly offer these clumsy tales (with their even clumsier illustrations) in the hopes that they might spark a few insights or arguments.
After all, insights and arguments are what math is all about.
Now, without further ado, I give you:
Chapter 1: The Bear in the Moonlight
It was a half-moon that night. The student and the teacher could see a shadowy, white-chested figure lumbering down the mountain path.
“Is that a bear?” the student gasped.
The teacher nodded calmly. “It may be. Or, it may be one of the children from the village, disguised as a bear, hoping to scare his friends.”
“Well, which is it?” the student hissed. “A deadly bear, or an innocent child?”
“Let us each determine the probability that the figure is a bear,” the teacher said. “Then we shall share our answers with one another.”
After a pause, the student whispered her answer. “20%. It could be a bear. But it looks too short, and I think it’s wearing a backpack.”
“Very good,” the teacher said. “I say 40%. It moves slowly for a bear, but it seems to me the right size.”
“So I’m wrong,” the student said. “It’s 40%.”
“No,” the teacher replied. “You are perfectly right. For me, it is 40%, and for you, 20%.”
“But you’re the teacher. You know more.”
“And your eyes are sharper than mine. Our perspectives are different, but neither is truer. I am right, and so are you.”
“So is it a bear,” the student said, with straining patience, “or not?”
The teacher closed her eyes. “What you seek is certainty. But a probability is only a perspective. Tell me, does that creature know whether it’s a bear or not?”
“So for the creature itself, the probability must be 0% or 100%. It knows with certainty. You and I have our own perspectives, and thus our own probabilities.” The teacher paused. “Tell me, if there were a full moon tonight, what would we see?”
“It’d be bright,” the student said. “We could tell at a glance if that shadow is a bear.”
“And if it were a new moon, what would we see?”
“Nothing. Darkness. There would be no shadow at all.” The student paused. “We wouldn’t see the creature approaching, so we wouldn’t even be having this conversation.”
“Precisely. When the moon is full and bright, we know all. There is no need for probability. And when the moon is new and dark, we know nothing, not even enough to ask a question. In either case – total knowledge, or total ignorance – probability is useless.
“Probability is for the nights like these,” she continued. “It is for the nights of half-light. It is for the nights when we can make out a form, but cannot tell its precise shape. It is for nights when light and shadow mingle, when knowledge and ignorance share our thoughts. It is an expression of our uncertainty – no more, no less.”
“So you’re saying,” the student said, “a probability depends on what we know, and what we don’t know. And because you and I know different things, our probabilities are different.”
The teacher smiled. Looking back out the window, the student found that the figure—bear, child, whatever it was—had vanished.
Probabilists like to list three approaches to probability: the classical, the empirical, and the subjective.
“Classical” probability refers to chalkboard situations, like rolling dice and drawing cards, where we naively assume that a set of outcomes are equally likely. We take for granted, for example, that heads and tails are equally likely when flipping a coin—even though, in real life, a coin has microscopic imbalances, making one side slightly more likely than the other. Classical probability, then, is a purely theoretical game.
“Empirical” probability relies on real-world frequencies. For a simple example, if 32% of skateboarders have broken their noses while attempting a trick, we give you a 32% probability of the same.
“Subjective” probability aims to express uncertainty in our minds – and it’s much trickier to define. What does it mean to say that this shadow has a 20% chance of being a bear? There’s only one shadow. Or that the President has a 74% chance of winning reelection? There’s only one election. Or that tomorrow’s chance of rain is 30%? There’s only one tomorrow. Because we can’t repeat these events, it’s not obvious at first glance what such statements really mean.
But that doesn’t make subjective probability meaningless. Rather than striking at theoretical truth (like classical), or statistical truth (like empirical), subjective probability strikes at psychological truth. In fact, one could mount a plausible (if controversial) argument that all probability is subjective, because all probability ultimately expresses a perspective of uncertainty.
I believe that this subjective notion of probability (as a state of partial knowledge) leads to a powerful understanding of topics like conditioning and coincidences. We’ll see that in future weeks.
I’d like to thank my father, James Orlin, for providing some foundational ideas for these stories, as well as helpful feedback and conversations. Also for being one cool cat.