The fourth in a series of seven fables/lessons/meditations on probability.
“I’m flipping a coin,” the teacher said. “Tell me the probability that it comes up heads.”
“Is it a fair coin?” the student asked.
“Yes,” the teacher said. “I promise.”
“Then the probability is 50%.”
The student heard the plink of the flipped coin, and the slap as the teacher caught it. “Heads. Let’s play again.”
“50%,” the student said. The coin spun through the air.
“Heads again,” the teacher called. “Keep playing.”
“50%,” the student said. A coin has two sides, so the probability of heads is 1 in 2. Easy, she thought to herself.
“Heads again.”
The conversation continued. “Another heads,” the teacher called. “We’re up to 30. What’s the probability that the next one is heads?”
“You’ve gotten really lucky, but the probability of heads is still 50%.”
The teacher shook her head. “You are a fool indeed.”
“But that’s how coins work,” the student said. “You told me it’s a fair coin, so the probability is 50%. All these heads have just been a wild coincidence.”
“Tell me,” the teacher said. “What is the probability of 30 heads in a row?”
“Um… 1 in a billion, more or less.”
“And there’s no chance that I’m swindling you, is there?”
“Well…” the student said. “It’s very unlikely. Not only would you have to lie to me, but you’d need to be able to manipulate coin tosses. Maybe there’s a 1 in a million chance.”
“1 in a million is quite different from impossible,” the teacher said.
“I guess.” The student shrugged.
“Now,” the teacher continued, “what is the probability that the next flip will be heads?”
The student paused. “I’m not sure. I thought it was 50%… but the chance of your tricking me is still 1000 times more likely than that you’re doing it honestly.”
“So is it 50%, or not?”
“It’s not,” the student concluded. “That’s a swindler’s coin.”
The teacher smiled. “Very good.”
The student’s face began to grow hot. “So you were lying to me this whole time! Why would you do that? You said it was a fair coin, and I trusted you.”
“As you should,” the teacher said, “up to a point. When you witness something that defies all of your assumptions about the world, you must learn to question those assumptions. To do otherwise is to disappear inside your own head, and ignore the world of evidence knocking at your door. You become one of three things: A dreamer, a fool, or a stubborn theoretician. And I’ve never had much luck telling the three apart.”
“You could have just told me that,” the student said. “You didn’t have to trick me.”
The teacher laughed. “How could I teach you about falsehoods, if I spoke only truths?”
Further Thoughts An Explanatory Rant
Some probability texts ask a similar question: “If a fair coin is tossed 50 times, and comes up heads each time, what is the probability that it comes up heads on the 51st toss?” The “correct” answer is ½. A fair coin always has a probability ½ of coming up heads, because that’s how we define “fair.”
But guess what? If a coin comes up heads 50 times in a row—a 1-in-a-quadrillion event—then that ain’t no fair coin. The question could be paraphrased: “If I tell you a coin is fair, and then overwhelming evidence accumulates to the contrary, would you still believe me?” And the “correct” answer would be: “Yes, because I never reconsider my assumptions.”
For probability to be useful, it ought to stay anchored in practice. We shouldn’t cling to invalidated assumptions or now-obsolete frameworks. We shouldn’t keep telling ourselves the emperor is clothed.
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 crazy fox.
Brilliantly written! This teaches not just probability, but on why we need to think critically (ask questions) instead of blind faith that our assumptions are correct.
Thanks! Inspecting one’s assumptions is a huge part of any mathematical modeling.
It’s an information literacy lesson, to get in the mind of the test author or teacher or whoever. “Is this a test where the questions are sneaky, or is everything perfectly literal? Will this teacher appreciate my clever and accurate response, or should I do the problem as implied?”
You’re probably right about the function of that question. But if it doesn’t test anything beyond information literacy, I’m not sure it’s worth asking. There are other ways to teach students how to read the intent of a question.
Very amusing and thought-provoking! I agree that all too often students are taught to apply formulae in thoughtless, mechanical ways without thinking through the information they’ve been given. Good point there!
Agreed. This one doesn’t even rise to the level of formula–it’s really just a question of parroting back the right information.
“The teacher laughed. ‘How could I teach you about falsehoods, if I spoke only truths?'”
By mentioning falsehoods instead of using them. Saying: “This coin is fair” is lying to your student. Saying: “Assume that someone you trust implicitly tells you that the coin is fair” is not lying to your student. perhaps the former is more effective, but that’s a different matter.
Good posting though.
Yeah, my dad made the same complaint. Lying certainly makes for a better story; honesty probably makes for a better lesson.
Honest people do not resort to fabrications even to show that dishonesty exists. If teachers implement this particular lesson, it will destroy their reputation for credibility in exchange for a gimmicky demonstration of something that students can easily observe in their own lives. There are many liars out there. Placing one in the classroom is not the answer. A favourite that I used with my children when they were young was the trip to the dollar store. The toys that broke after one day of normal use helped them to understand that there is a great deal of value in approaching the market with skepticism. But what is being advocated here will throw the entire profession of teaching into disrepute if acted upon. Is it really advisable that students be unable to trust anyone, ever?
Agreed that teachers probably shouldn’t ever lie to their students. That’s why this is presented as a fictional story, not as a recommended lesson plan!
This reminds me of an encounter with one of our school’s boarding students. I was on duty driving kids to and from a movie theater. I had a van full of girls singing on their way home and it made me laugh. One of the girls asked why I was smiling and I said that their singing made me happy. She replied “Teacher – you should not lie to students.”
I had a teacher who promised to always tell us the truth. He went on to explain that sometimes the truth would be told after a lie, but that it was necessary. We knew that he had our best interests in mind, we knew that he would eventually tell us the truth, and we knew that we should always be ready to check things out for ourselves. I’m sure a few students decided not to trust him, but this helped all of us remember that he was not the source of all knowledge.
I am fairly certain these fables aren’t intended to be used as lesson plans, so the comments about lying seem to be missing the point. My guess is that most math teachers do tell lies. I know that many algebra 1 courses would have us tell students that an answer is “all numbers,” only to come back later and tell them that what we really meant was all REAL numbers. Maybe you’ve never done that, but I know that I warn my students that they will not always learn the whole truth until the end of things.
And of course, while I was composing this the author said as much about these NOT being lesson plans.
Well, I’m glad you chimed in. You’re right that often in math (and almost constantly in science) teachers give first-order approximations of the truth, which are, strictly speaking, lies. If you want to teach kids about, say, matter, you start with a simplified model of an atom, not with cutting-edge research on the Higgs Boson. As Emily Dickinson put it: “Tell all the truth, but tell it slant.”
It could also be a coin with two heads.
Flipping maybe would be fair, depending on your definition 🙂
True. It’s equally likely to land on either side, after all. 😉
A wonderful story line, I really liked the implications here, I haven’t tried this before, but it has me thinking. Great stuff.
wow this post should be on some internet “best of” list somewhere. brilliant way to look at probability.
well, this comment definitely goes on my “best of” flattering comments list, so thanks
“If a coin comes up heads 50 times in a row—a 1-in-a-quadrillion event—then that ain’t no fair coin.” – no, it most probably isn’t, but you can’t be certain. Isn’t that being guilty of the assumption you’re quite rightly highlighting.
Well, this assumption is really more of a conclusion, emerging from an overwhelming preponderance of data.
Technically, you’re right. We can never be truly certain about anything. But rather than insert “probably” or “most likely” into every single one of our statements (“Columbus most likely sailed the ocean blue/ Probably in 1492…”), I think it’s okay to make straight-up assertions in cases of near-certainty (like this one).
This post makes me think of two things that I share with my stats students. The first is the opening scene of Tom Stoppard’s play Rosencrantz and Guildenstern Are Dead. The second is a lovely experiment in testing people’s sense of randomness. There is a stats professor who asks a group of students to toss a coin 100 times and record their results. The other group of students are asked to imagine tossing a coin 100 times and record their results. She can – with a high degree of accuracy – look at the results and determine which group really tossed coins. It is not unusual in a string of 100 to see runs of 5, 6, or even more of the same result. When imagining the results students will rarely have more than two or three of the same result in a row. It does not ‘feel’ random at that point.
Another terrific post, thanks!
Ooh, I need to go back and re-read (or re-watch) R&G are Dead. Saw a stage performance in high school and really loved it.
As for the other experiment: That’s a favorite of mine, too. In fact, it shapes the upcoming 7th-and-final story in this series!
This proves that I’m a psychic, right?
Well… either you are or you aren’t, so I’d say there’s a 50% chance. 😉
Hadn’t thought about the fallacy of that type of question before. Reminds me of high school physics. Assume there’s no friction, assume an ideal spring, assume no air resistance….
Yeah, economists also do this–“assume perfect information, assume well-ordered preferences…”
Hey Ben,
I’m planning an informal talk to a group of my teenage peers at a University Maths Club, and I’m doing it on probability theory. I love the little stories you’ve written, and I’d like to start by reading one or two of them to the group.
I can’t seem to find your email to ask you if I can use them, so I’ve written this comment. I can give you an email address if that would be a preferred method of communication.
Thanks,
Ben
Hey Ben, feel free to use them! (My email is just my name at gmail for future reference.)
Thanks muchly 😉
I have taught some computer science classes, and I always say “There are times when I will lie to you, to simplify a concept. Then later, we will learn the complex parts that expose my lie. So when I lie to explain something in simple terms, I will do this (show crossed fingers) – so you know that this is a truth in the current context, but will be broken later.” This helps the immediate understanding, and the understanding that this isn’t ALL the truth ont he current subject. It also helps keep the smart-asses quiet 🙂
You have no idea how glad I am you wrote this. This is EXACTLY what I think every time I hear someone say that.
(“That” being “The probability of a coin coming up heads on the 51st flip for the 51st time is 50%”.)
If you replace your “stubborn theoretician” with “economist” at the end, the whole still works fine.