The Incalculable Joy of Fermi Questions

For me, life is just a series of Fermi questions.

“How big is the book publishing market in the U.S.?” my friend John asked me as we strolled through the Harvard Coop.

I remembered reading somewhere that the average American buys 5 books per year. Or maybe I made it up; either way, seems about right. Call that $100 per capita. With 300 million of us, that’s a $30 billion market.

(Actual number: $26 billion.)

“Guess how much blood a 16-week-old fetus pumps per day?” my wife quizzed me.

Well, I know an adult has 8 liters of blood. Say that it circulates once per minute. That’s 1500ish times per day. Hence, 12,000 liters per day. But a fetus is perhaps 1/10th the height, so its blood volume is 1/1000, giving us 12 liters per day. Then again, their heart-rate is twice as fast, so let’s double that to 24 liters per day.

(Actual number: 25 quarts. Basically identical.)

You might know them as “Google interview questions”; those funky estimation problems that you’d never know off the top of your head, but to which you can reason your way.

Whatever you call these exercises, I find them inordinately fun.

2018.8.21 meta-fermi

Since Fermi questions are so fun and useful, why aren’t they more widely taught? Why isn’t every middle school student doing one of these per week all year long?

I suspect a prosaic reason: they’re hard to grade fairly. Math education is accustomed to cut-and-dry answers. Fermi work is more like an essay, where there are many plausible answers, and reasoning trumps conclusions.

All the more reason to embrace them, I say!

2018.3.15 bond villain fermi questions.jpg

11 thoughts on “The Incalculable Joy of Fermi Questions

  1. Which all goes my *my* favourite topic: context. May I politely suggest that it is irrelevant to throw numbers about, such as, “People, on average, buy $100 worth of books per year.” When the cost of a single text book is over $120; when the market for resold paperbacks may be $2 books, when the content of each book is so much more important to consider – unread mandatory texts, gardening or other “how to” books, novels, poetry, graphic novels, bibles…
    Ben, what addition to the body of knowledge is made by making up “averages”?
    But I still like your blog 🙂
    All the best of the holidays!
    George

    1. The use of averages here is to plausibly answer the question posed: “How big is the book publishing market in the US?” Of course, one might have the answer of $26 billion available and there’s no need to do a back-of-the-envelope estimate. However, when such (refined) data is unavailable, it is useful to use averages for efficiency and relatively good approximation. The procedure Ben outlines tends to work because the over and under estimates “tend” to cancel out. It’s somewhat analogous to the central limit theorem of probability. And this Fermi-question-type approximation is done quite often by companies that need to estimate costs or number of units of a product but do not have the time or resources to conduct market research.

      More refined data can also be irrelevant for the final estimate. For example, the cost of textbooks and paperbacks is not immediately relevant since the question is not asking about the textbook publishing market nor the paperback publishing market.

  2. Thanks–never heard of “Fermi question” per se. Googled “What Are Fermi Questions?”. Got a nice list. Onward to fun and challenge.

  3. Ben, you must promise us (not that it will happen any time soon at all), that when you do eventually depart this Earthly realm you will leave your brain to science, on the off-chance it can be unravelled. Thanks!

  4. I’ve been teaching Physics, and usually do a Fermi Question project at the end of the year. They have to present their project to the class.
    https://docs.google.com/document/d/11YKZDbs11XQr2AGsImz0rUoQoP5v4EjyBJbsDxFLt-4/edit?usp=sharing

    The best question EVER was from this quiet girl… “If you killed every human on Earth and tied their intestines together to make a rope, how long would it be?”

    This girl got up on front of the class, stone-faced, and discussed murdering everyone… and she had the BEST researched project of the year.

    She took her shoelaces and measured how much length would be lost in each knot (multiplied by the number of humans = number of knots).

    She had a plan for how the last person would end the rope (start seppuku, cut their intestine, pull one end out, tie it to the rope, then run backwards, stumbling to their death as they pulled the end of the rope taut).

    The class was shocked as she talked about how she estimated the length of babies intestines, but not fetuses.

    The quality of her research was top-notch, her estimates were well-reasoned, and I was impressed. Creeped out, but impressed.

      1. By all means, I stole it from someone else.
        I use it for “dead time” like Finals week, when its hard to start something new. Keeps kids active, interested, etc.
        There are some good YouTube videos introducing the idea and basic strategy of Fermi Questions, I show one to introduce the idea, then turn them loose to make mistakes and learn from them.
        Easily one of the more popular projects we do.

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