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?”
“Of course.”
“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.
Further Thoughts
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.
Get the pdf, or check out other stories in the series!
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.
I like your bear story. Non-mathematical entrances into mathematical concepts can be really wonderful. I can’t wait for the rest of the series.
The two things that I like to share about probability with non-math people is first that Newton did exactly what your character suggested – he invented tools for looking in the light (calculus and a telescope). He, and those who folowed, explored just about as far as you can go by looking into the light (motion of the planets, classical mechanics, etc…). When you are done looking into the light, all that you have left is the other 99% of the universe that is not in the light (this is probability, uncertainty…).
Second, I like to talk with people about the wonderful contradiction that a subjective science can be so simply distilled into a few elementary rules (Kolmogorov’s axioms).
Thanks for reading! Those are both great anecdotes about probability.
I think the telescope predates Newton, but I really like the idea of Newton pushing past the light into the vast, unknown darkness.
And weirdly enough, I’d never heard the phrase “Kolmogorov’s axioms,” even though I’ve obviously taught them to probability students. So thanks for the reference!
I think that what Newton did do is to make the first reflecting telescope, based upon the ideas of others. More importantly, he invented the method by which we still make spherical lenses to this very day, simply rubbing two discs together. I just did a very quick google of this, and can’t find any good reference online, so I will have to look it up in an actual book.
What I like about Newton is that he made all of his discoveries as a young person, and was as clever at building physical things as he was at building mathematical things.
What I like about Kolmororov’s axioms is that they are not too abstract and removed from how we study probability. A simplified version can be presented to kids, without generating too much resistance.
Agreed on Kolmororov. In fact, it’s a very clean way to (re)-introduce high school students to the idea of an axiomatic mathematical framework.
Cool stuff about Newton–didn’t know that about the spherical lenses. I love the Neil Degrasse Tyson take on Newton: http://www.youtube.com/watch?v=danYFxGnFxQ
I am not a Mathematician nor a native speaker of the English Language. But I love learning and teaching love for knowledge, I am an educator in a Latin American country. I used to hate Math, or teaching Math. But thanks to blogs like this and Dan Meyer’s, I am finally grasping gradually how to understand and appreciate it better. Thanks for your efforts to translate number into words, stories… I still don’t understand many concepts well. Before I didn’t care if I couldn’t figure it out correctly, but at least now I enjoy trying to understand it!!!
It’s great to hear from you! My hope is to give a different, perhaps more engaging take on mathematical ideas, so I’m really glad you’re enjoying it. If I can ask–what country are you in?
Your comment about concepts being smooth and hard to grasp is spot on! Sometimes you can “feel” a concept in your brain – like feeling a watermelon seed slip through your fingers – but have to work really hard to contain it. Thanks for adding some texture to probability. I’m looking forward to the future posts.
Thanks! It’s an experience I’ve had myself, and one I see all the time with my students. You can see it on their faces when there’s an idea dancing at the edge of their understanding, but they can’t quite pin it down.
Great article! Would you consider something like the Monty Hall problem a good introduction as well?
Mmm, good question. I go back and forth.
Monty Hall seems like a great problem for maybe 2-3 weeks into a probability unit. It hinges on a subtle understanding Monty’s door-choosing behavior: he will only open doors that (a) you did not pick and (b) do not have the prize. In light of the complexities, it might work better as a (scaffolded) chance to practice problem-solving skills than as a way to acquire them to begin with.
What are your feelings?
That makes sense. I used to introduce Monty Hall pretty early in my probability unit, but I didn’t think my students had a great understanding of the logic. I think most adults have a pretty hard time with understanding subjective probabilities – the scenario you give where tomorrow’s chance of rain is 30%, for example, is one most people have trouble wrapping their heads around.
That said, I see starting students with subjective game probabilities as an effective way to demonstrate the power of probabilistic reasoning. If I can show students that my knowledge of probability can give me a significant advantage at card games, board games, or contrived games such as Monty Hall, I can hopefully motivate their interest in the subject. Even if they can’t work those probabilities to their advantage early in the unit, the possibility that they could eventually do so can hopefully ignite a passion for learning the topic.
That seems like a good approach: building motivation and interest by playing around with games, and later building up some abstractions.
Agreed that the “30% chance of rain” is kind of a weird example. I usually explain it by starting with a frequency explanation (“On 30% of past days like this, it has rained”) and then building to subjective (“What does ‘like this’ really mean, since no two days are precisely alike? The forecasters must be making some judgment calls about which features of today’s weather are relevant for predicting tomorrow”).
Wish I’d found this sooner so I wouldn’t be so late to the game.
My take on the 2 door problem is that most who point to the mathematical solution seem to assume that the door that is shown was picked randomly. If we watched a bunch of reruns we could presumably get a big enough sample to determine an empirical probability and see how it correlates with the purely theoretical probability. Inducing a contestant to change their choice is good publicity if they win. If they lose it reduces expenses. Perhaps that means there’s no incentive for Monty to game the system by opening a door more often when the contestant has picked the correct door, but I don’t know. I’m happy with the subjective probability that if there’s 1 good prize and 3 doors there’s a 1 in 3 chance I’ve picked the right door. Whether or not Monty opens one of the other doors doesn’t matter.
As for the 30% chance of rain, I had a coworker who thought that meant it would rain in 30% of the areas the forecast applied to. He seemed unable to grasp that he was describing a 100% chance that it would rain somewhere.
This is beautiful. That’s all I can say.
Thanks for reading! 🙂
Do you ever have trouble teaching students to distinguish between the “truth” of the subjective probability vs. the actual truth of the fact. In the bear/child instance, as you noted, the thing is either a bear or a child (assuming you have 100% chance that it is actually a thing). Yes the decision making of the two onlookers can and probably should be guided by their judgement of the probability that the thing should initiate fight or flight. But do many people start to create theories of subjective reality based on statements like: “No,” the teacher replied. “You are perfectly right. For me, it is 40%, and for you, 20%.”? It, in this case is the probability not the fact but what if that pronoun is switched for the bear/child entity in question, the statement then becomes a fallacy but one that far too many people fall into? I’m not saying it’s wrong just how do you avoid the fallacy in your teaching?
That’s a good question.
I try to emphasize that a probability is a perspective. A probability can be “right” or “wrong” only insofar as the person uses (or fails to use) all the information at their disposal.
Another example that perhaps illustrates it more clearly: You flip a coin, look at it, and ask me what the probability is that it’s heads. The best answer I can give is 50%, even though YOU know what it is with certainty. It’s not that you and I exist in different realities; we just have different levels of knowledge about the reality that we share.
I’ve only taught this material for a couple of years, but in my experience students aren’t eager to relinquish their belief in objective reality, so not too many fall into the misunderstanding you’re describing (though I can see the potential danger).
I like the coin flip one because it takes out nearly all possible ways to confuse actual reality with perception. The reason I asked the question is that in a Philosophy class you might find a very different reaction to questions about objective reality. There are so many things we humans are learning now particularly at super small and super large frames of reference. I have had many conversations with people who are very smart people but who are convinced that what we are seeing indicate that there are multiple objective realities when it looks to me that they are just seeing very sophisticated versions of the bear/child. Maybe someday I’ll be proven wrong but I don’t logically see how that can occur. I can’t get past the axiom that there is a reality or A is in fact A and not just sometimes A and sometimes not A. Anyway because of those conversations I’m probably overly sensitive to statements like “for me it’s this and for you it’s that” (PS: I’m also very skeptical of pronouns because a pronoun must be related to a previously used noun and the possibility of the speaker relating it to one noun and the listener relating it to another is higher than most people would like to think).
This is a fun Blog, thanks.
Yeah, those are interesting thoughts. I’m sure there are philosophically consistent worldviews that deny the existence of one objective reality, but I’m also sure such a worldview would come with loads of consequences and implications I would find ludicrous or impossible live by.
Your point about pronoun confusion is especially applicable in pop songs, where the “I” and the “you” are deliberately vague, to invite people to insert themselves!
I start probability in my AP Statistics class next week, and am really stuck. I’m so glad someone shared this with me!
My students come into my class having learning “probability” for years, and they’re totally ready for me to break out some red and blue marbles. I’ve been struggling with where the disconnect happens, and I definitely plan on sharing these stories with my students. Thanks for your great work!
Also, is it ok to post these stories on my class blog? (Its subscription only for students in my AP classes.)
Sure–let me know how it goes! I haven’t had a chance to use this in the classroom yet, and I’d love to know how your kids respond.
I will! I’m still working out my ideas, but I’m posting your story on my class blog on Friday–we’ll be doing some AP prep (especially around multiple choice) early next week before probability, but I want to weave in some thinking about probability before we really work with it, and I can’t wait to try the stories. Here’s where I’m at so far: http://xypi.wordpress.com/2013/10/31/lions-and-tigers-and-bears-rambling-about-probability-again/