On Fermi Calculations

Big Falling Fermi Confetti

Take a piece of standard paper. Rip it up, roughly, into an 8 by 8 grid, which will net you 64 pieces of over-sized confetti. Hold them in your cupped hands at chest height, and let them fall to the ground.

What can you learn from doing this? Well, for one thing, it can help you estimate the strength of Trinity, the first atom bomb test explosion.

No, really, it can.

But I’ve left out two things. You have to have been at the observing site at the time when Trinity was actually detonated. And you have to be Enrico Fermi.

Enrico Fermi 311x300

Enrico Fermi in front of a machine he invented for toasting bread.

Fermi was born in 1901 in Rome. His was the trajectory common for physics geniuses. Graduated top of his high school, took first place in his university entrance exam, was acknowledged to be “unteachable” by his physics prof (not that he was dumb; there was just nothing new to teach him, as he already knew it all), and so on. When 19 he started work on x-ray crystallography, and in his third year was already publishing papers. By the age of 24 he was appointed professor at Sapienza University in Rome. He won the Nobel Prize in Physics at the age of 37.

Brilliant in both theoretical and practical physics, Fermi was one of the early experimenters in nuclear fission, and designed Chicago Pile-1, the first reactor to achieve a self-sustaining chain reaction. He was not slow to see the dangers of this technology, and advised the US military on its potential uses. Eventually he worked on the atom bomb at Los Alamos, and observed the Trinity Test on 16 July, 1945.

It was here he did his famous paper test: by pacing off how far the paper confetti was blown by the blast wave, guessing the weight of each piece of paper, and doing a few back-of-the-envelope calculations, he estimated the yield as 10 kilotons of TNT. The currently accepted value is between 18 and 19 kilotons.

That in fact is astonishingly accurate, given what he had to work with, which was basically nothing. Don’t forget, this was the first-ever blast, and this was the age when the word “computer” meant a human being who did calculations as a member of a team, with pencil and paper or a mechanical adder. (Although it must be said there were a very few electronic computers at the time, and the Los Alamos project did use them.)

Two Wrongs Can Make A Right

Throughout his career, Fermi was noted for his ability to produce reasonable guesstimates in situations in which he had little or no solid data. LIke the confetti test. His basic method was to break a problem down into many constituent parts, and estimate values for each. By adopting this strategy, a thinker benefits in two ways:

First, the component problems frequently can be estimated with more accuracy than just a single shot at the larger whole. If you have no idea of what the Trinity yield might be, you can guess the weight and surface area of your confetti, pace off the distance, know how far you are from the blast, and so on.

Second, in the absence of any systematic bias, over- and under-estimates of all components tend to cancel each other out. In curious contradiction to the adage “two wrongs don’t make a right” (stolen, I would admit, from the world of home spun ethics, not physics), you can home in on the correct value (with a bit of luck).

Fermi became so good at this technique, that today it bears his name. We call them Fermi Calculations or Fermi Problems.

Let’s take an example.

VWs Sold In Germany

A while back I was in Frankfurt, visiting a very good friend, and somehow I happened to mention Fermi Calculations. She hadn’t heard of them before, I explained, and she immediately wanted to have a go. So we set ourselves the problem of trying to calculate how many Volkswagens are sold every year in Germany. We were in my VW Tiguan at the the time, so had no decent access to the Internet. Thus, we were forced to guess every step of the way.

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Volkswagen. Vorsprung durch betrügt Emissionstests Technik

There are many ways of approaching such a task; we tried a top-down strategy, starting with the overall population of Germany. As I cannot remember exactly how our guesses went, I can only supply a rough reconstruction of our steps, which appears below. Real answers, since researched where possible via Wolfram|Alpha, Wikipedia and other web sites, follow our estimates, in brackets:

  1. How many people live in Germany? We guessed 70 million. (81.6 million is the current 2014 estimate, so we were significantly low on that one.)
  2. What is the ratio of cars per person? We broke this down a little bit. We reckoned most single people own a car. Maybe close to 50% of families own one car, the other near 50% own two, and only a negligible number of families own more than two cars. Perhaps we should have attempted to estimate the ratio of single people to families, but in the end we simply guessed one car for every two people, which would be about 35 million cars in Germany, based on our 70 million population estimate.
  3. But of course this leaves out a lot of business vehicles. VW don’t actually do fully-fledged trucks, but they do vans and pickups. There are far fewer of these than cars, and we didn’t see any easy way to break this estimate down any further. So we threw in 5 million more vans and pickups, to arrive at a total number of 40 million vehicles. (True values: cars 46.6 million, vans and pickups 2.6, for a total of 49.2 million vehicles. So we were low on cars, high on vans, and low on the overall total).
  4. Assuming the total number of vehicles in Germany stays roughly the same year in, year out, we decided to have a wild stab at guessing the average vehicle life-expectancy. I recall we guessed this was 10 years (which seems high to me now, but I do believe that was the figure we used). So in order to replace those dying vehicles, 4 million new cars need to be sold each year. (According to Statista, the number of vehicle sales in Germany in 2015 was 3.5 million http://www.statista.com/statistics/265951/vehicle-sales-in-germany/. So we were pretty close, but a bit high.)
  5. Guessing the proportion of VWs to other brands in Germany was not an easy task. The country would appear to be a nation of nationalistic car-buyers, with German cars far outweighing anything else. In some place comparatively rich like Frankfurt (where we were), you see a lot of higher-end Mercedes, BMWs and Audis. In less-wealthy areas, cheaper cars predominate, and VWs really come into their own. So we guessed VWs made up one fifth of the total car population, nationwide. So, in order to keep VW’s market share roughly stable over time, they have to sell one-fifth of our 4 million new cars estimate, or 800,000 vehicles per annum. And that was our final guess.
  6. The web site Best Selling Cars (http://www.best-selling-cars.com/germany/2014-full-year-germany-best-selling-car-manufactures-brands/) states that the Volkswagen brand had 656,494 vehicle registrations in 2014.

So we got pretty close. As I stated earlier, I cannot remember exactly each estimate we came up with in every step along the way (the exercise occurred over a year ago), but this does present a fair reconstruction.

To be clear, I am not trying to claim we are brilliant estimators; quite the opposite. What is interesting about the exercise is it reveals how breaking the problem down genuinely does make it tractable. Without the breakdown, I would have had no idea how many cars VW sold in a year. Any estimate would have been a complete stab in the dark. On the other hand, I had a certain amount of confidence that each of our individual steps were at least reasonable. Add up a bunch of reasonables, and, with a bit of luck benefit from under- and over-estimates canceling each other out, you end up at a new reasonable.

There’s More Than One Way to Skin An Estimate

In fact, before we were able to check our first calculation via the Internet, we decided to attack the problem in a completely different way. Our first approach was clearly top-down, starting with the population of Germany. We tried bottom-up: My friend lives in Bad Homburg, which boasts one VW dealer, albeit fairly large. I had recently bought the car we were driving around in, in Albertville, France, from a dealer about the same size. I had chanced then to ask him how many cars he sold in a year, and he told me. We used that as an average number of cars sold per normal-sized dealership.

Although it is true the French don’t have nearly the proportion of German cars on their roads (they too, are nationalistic in their buying decisions), we decided a dealership count would adjust for this difference all by itself: Germany would have more German car dealers than France. So we then guessed the number of towns in Germany of similar in size to Bad Homburg. We thought for a while about how you could adjust this to arrive at an overall national VW dealership count by taking into consideration the many-fewer-but-larger towns, and the many-more-but-smaller towns in the country. By multiplying our estimate of cars sold per dealership by dealerships per town by towns in the country (adjusting as we went), we eventually arrived at a figure that happened to be pretty close to our first estimate of 800,000.

I have to admit it is very possible, even probable, that we were unconsciously adjusting our intermediate calculations so we arrived at an agreement with our first estimate. Daniel Kahneman would call it a kind of constructive confirmation bias. Having said that, at the time we did our second calculation, we hadn’t yet confirmed the first. So at least, if we were unconsciously cheating, we were favouring a figure we didn’t yet know was fairly accurate.

But (assuming the current reader cares to give us the benefit of the doubt), it is gratifying that both methods got us pretty close to the true answer. Once again, a bunch of small reasonables can add up to one large reasonable.

Yeah, But It Is Easier to Look It Up….

Of course in this web-enabled age, it is much easier — not to mention more accurate — simply to look things up. Why take the stairs when there’s an escalator?

Curiously, the answer to this question is pretty much the same for Fermi Calculations as it is for getting around in an airport or shopping mall: it is good mental exercise, it is challenging and educational to break a problem down, and it is fun.

Another point, no doubt already evident to the current reader, is that the whole process is immensely satisfying when you get it even near right. We were both beaming when we looked up the true answer and discovered we’d come pretty close.

And finally: we do all have the odd, occasional (hopefully non-nuclear) Trinity moments, when a question arises that simply can’t be answered by Wolfram|Alpha or a Google search. Who ya gonna call when even the Web doesn’t know? Your Fermi calculation skills. Because,

There are more things in heaven and earth, Horatio,
Than are dreamt of in your Internet.

VW Logo 40x40

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