How many piano tuners are there in Chicago?
One of the chapters in Crisis of Control is on the Fermi Paradox, a fiendishly simply-stated problem with existential ramifications. That kind of simplification of the complex was the stock-in-trade of physicist Enrico Fermi, a man who could toss scraps of paper into the air when the atomic bomb test exploded and calculate in seconds an estimate of its yield that rivaled the official figures released days later. He taught his students to think the same way with this question: “How many piano tuners are there in Chicago?” No Googling. No reference books. Do your best with what you know. Go.
This is one of those questions where “Show your work” is the only possible way to evaluate the answer. The lazy ones will throw a dart at a mental board and say, “X,” and when asked how come, shrug. The way to solve this is to break it down into an equation containing factors that can be more readily estimated. If we knew:
- P – The population of Chicago
- f – The number of pianos per person
- t – The number of times a piano is tuned per year
- H – The number of hours it takes to tune a piano
- W – The number of hours per year a piano tuner works
then the number of piano tuners in Chicago is: P * f * t * H / W . Here, let’s walk through this:
- P * f gives the number of pianos in Chicago, call that N. P and f are each easier to estimate than how many pianos there are in a city.
- N * t gives the number of piano tunings per year in Chicago, call that T.
- T * H gives the number of hours spent tuning pianos per year in Chicago, call that Y.
- Y / W gives the number of piano tuners it takes to provide that service. QED.
Of course, you could look at those figures and say, wait, I don’t even know the population of Chicago, much less how many hours a piano tuner works. But it’s easier to make a good guess. To get f, you can go off your personal experience of how many friends’ houses you’ve seen with pianos, make a correction for the number of pianos in institutions of some kind (theaters, schools, etc), and at each stage, add in confidence limits of how far off you think you could be.
This process is what leads to the most important math in the Fermi Paradox chapter in Crisis, the Drake Equation:
N = R* · fₚ · nₑ · fₗ · fᵢ · fᶜ · L
where:
N = The number of civilizations in the Milky Way galaxy
(ours) whose electromagnetic emissions are detectable
(i.e., planets inhabited by aliens sending radio signals)
R* = The rate of formation of stars suitable for the
development of intelligent life
fₚ = The fraction of those stars with planetary systems
nₑ = The number of planets, per solar system, with an
environment suitable for life
fₗ = The fraction of suitable planets on which life actually
appears
fᵢ = The fraction of life-bearing planets on which
intelligent life emerges
fᶜ = The fraction of civilizations that develop a
technology that releases detectable signs of their
existence into space
L = The length of time such civilizations release
detectable signals into space
And that gives us a way of estimating how many intelligent civilizations there are in the galaxy right now, from quantities that we can estimate or measure independently. Of course, the big question is, why haven’t we found any such civilizations yet when the calculations suggest N should be much larger than 1? But NASA thinks it won’t be too long before that happens. And when we find them we can ask them how many piano tuners they have.
