Basic Concepts: Fermi Questions

Closely related to the idea of order-of-magnitude estimates is the idea of Fermi Questions, a type of problem that demonstrates the power of estimation techniques. The idea is that you can come up with a reasonable guess at an answer for a difficult question by using some really basic reasoning, and a few facts here and there.

The classic example of a Fermi Question is "How many piano tuners are there in Chicago?" This has never really worked for me, though, because I don't know anything about pianos, and I have no real way of knowing how often they're tuned, or any of the other estimates. So, to demonstrate the basic idea, I'll make up a different question:

How many shopping malls are there in the United States?

It's not immediately obvious how to attack this, but a reasonable assumption might be to say that a mall needs a certain number of customers within reasonable driving distance in order to remain open. If the population is too low, you won't get enough shoppers to keep a mall open.

So, how many shoppers does it take to keep a mall open? Well, within easy driving distance of my house (~30 minutes), I can think of five malls, so this area must contain at least the minimum number for five malls.

The population of this general area is probably between 100,000 and 200,000 people-- Schenectady and Troy are probably 20-30,000 each, Albany proper is probably 40-50,000, and the various suburbs account for a good number more. Let's call it 150,000, because the math works out nicely that way.

Five malls for 150,000 people works out to 30,000 customers per mall, if we assume they're just barely keeping the malls open (a reasonable assumption for a couple of those). The total population of the United States is something like 300 million people, so we would expect 300,000,000/30,000 = 10,000 malls in the United States.

This happens to be a question that's susceptible to Googling (which is why I picked it, though I didn't look at the answer in advance), and a few seconds of searching turns up a Census Department pagefrom 2005 claiming that the number of malls and shopping centers was 47,835 in 2004. So, at the very worst, the Fermi question reasoning gets within a factor of five of the correct answer, using very little concrete information. And that statistic includes "shopping centers," however you define those, so the number of malls meeting my mental definition is probably closer to the estimate of 10,000.

This is somewhere between a party trick and an essential physics technique. If you're thinking about doing an experiment in physics, you almost always start off with this sort of reasoning about what sort of signal you can expect. It's a good way to avoid hours and days of wasted effort, not to mention wasted money. And with a little knowledge of basic physics, you can get remarkably close to a lot of answers.

Of course, it's also a fun way to kill time, in a nerdy sort of way...

If you know of a particularly entertaining Fermi Question, leave it in the comments.

More like this

Starting with the January 1983 (v51, #1) issue of the American Journal of Physics, and continuing for a few years, Edward Purcell wrote a "The Back of the Envelope" column, which featured 3 such problems each month, with his solutions to the previous month. For the first installment, he wrote a "round-number handbook of physics," instead of solutions.

One such problem: "Rubber gloved, you are dribbling on an infinite court a basketball that is charged to 10kV. How much energy is emitted as electromagnetic radiation per bounce?"

Sanjoy Mahajan's text here poses quite a few.

How far can birds (and 747s) fly without eating? Why are raindrops a few millimeters in radius? How high can animals jump? How tall can mountains grow? How cold is the air at the top of Mt Everest? How hot is the interior of the sun? How much energy do gravitational waves carry? How fast do tsunamis travel? Even when these questions have exact answers, they are buried in the solution of complicated, often nonlinear differential equations. But by skillful lying -- the art of approximation -- you can understand these and other phenomena, and can enjoy the physics, chemistry, biology, and engineering of the world around you.

My masters qualifier had a question which involved some more technical sorts of estimates, but it was to estimate the neutrino flux thru your body from the nuclear power plant located a few miles across the river.

I nailed it.

On average, how much rubber wears off of your automobile tires each time the wheels make one turn? How much thinner is the tread per revolution?

Does it take many wheel revolutions to remove a layer that is one rubber-molecule thick?

Or do you remove about a one-rubber-molecule thick layer per turn?

Or do you remove a layer that is many molecules thick on each revolution?

By Emory Kimbrough (not verified) on 18 May 2007 #permalink

These types of questions are also common job interview questions in my line of work, software development. They were popularized by Microsoft and Amazon and now most competent interviewers use them as part of the screening process.

For me, I couldn't care less about the answer a prospect gives. I just want to hear her thought process and the mental leaps that went into deriving an answer.

By Brian Mars (not verified) on 18 May 2007 #permalink

One that seems popular here for physics undergrads is how many atoms are added to a human hair per second.

I got that one on a non-calculator scholarship test, and then again in our first advisor group tutorial once I arrived (at a different university). That tutorial included a range of others as well, but it is always the one that with me.

By Paul Schofield (not verified) on 19 May 2007 #permalink

How many times have The Eagles played Hotel California?
(Substitute your own band/song if you know nothing about the history of The Eagles.)

By Pseudonym (not verified) on 19 May 2007 #permalink

If you're thinking about doing an experiment in physics, you almost always start off with this sort of reasoning about what sort of signal you can expect.

There is something I was taught as "Wheeler's First Moral Principle", which is that you should not try to compute the answer to a problem without knowing (in the sense of a Fermi question) the answer in advance. Same as your example, which is why Fermi made those questions the centerpiece of his seminars. You can read some examples in the "Los Alamos Primer" where they figured out what was needed for the Manhattan project.

That human hair question is, IIRC, from a freshman textbook. There is also a book of harder problems (like that nice one from Purcell) called "Thinking like a Physicist".

One interesting question is to compare the odds of becoming a physics professor if you take a HS physics class (like PSSC, not some "physical science" course) to getting into the NBA if you play on a varsity HS basketball team.

By CCPhysicist (not verified) on 19 May 2007 #permalink

There was a physics class this year at Caltech, Order of Magnitude Physics, that was all about this. The lectures and problems sets are on the web page. Some of the problems anyone with HS physics could take a good shot at, but a lot of them need some physics background. I heard it wasn't a particularly easy class.

One particularly memorable problem I heard from a phys major friend was: "How large can the Jolly Green Giant grow and sustain himself solely on photosynthesis?"

A problem of definitions:

I'm pretty certain that the Census's definition of "malls and shopping centers" reaches pretty far down into "strip mall" territory.

So your estimate of "10,000 malls" is probably closer to the reality of what you were asking than is the Stat Abstract's answer of ~48,000.

(Oh, and I happen to work with the demographic data for NYS: your population estimates were low. Just for the record - Albany just dropped under 100k, Schenectady and Troy are at about 50k each; our entire Standard Metropolitan Statistical Area (which goes out quite a ways, and includes Saratoga) is probably closer to 900,000 than it is to 150k.)

(I'll stop now. But scratch a nerd and we bleed numbers.)

By Bob Oldendorf (not verified) on 20 May 2007 #permalink

My instructor for the equivalent of high school physics back at community college love Fermi problems and at one point gave a couple as bonus problems on a midterm. He gave a relatively straightforward one that I don't remember anymore, then threw us "What's the average velocity of all the automobiles in the world right now?" Most of us fell into the trap, coming up with an estimate of the average speed rather than realizing that velocity was a vectorial quantity and we were making the problem way harder than it needed to be.