## How Many Piano Tuners in Chicago?

The physicist Enrico Fermi was fond of asking this question in his lectures to students in Chicago, to show how most people can come up with answers to questions about which they have no expert knowledge, if they make rough guesses based on ordinary assumptions.

a whole range of scientific questions can be answered with very little basic knowledge

He showed how a whole range of scientific questions could be answered with very little basic knowledge – guesswork really – along with one simple rule. With his piano-tuner question, he’d show how the roughest guesses about population size, the proportion of people with pianos, how often pianos require tuning and so on, can be used to arrive at an approximate answer. He then showed students that they could use similar guesswork to arrive at useful answers to questions about science, such as:
⇒ What is the mass of the Earth?
⇒ Who walks faster – short people or tall people?
⇒ How much food goes into physical work and how much into keeping you alive?
⇒ What’s the total mass of all the students in your school or co-workers in your office?
⇒ How many cells are there in the human body?

These questions have become known as ‘Fermi questions’, and they are often used in teaching.

## An Example

Question : How many years of life does the average smoker lose?

So does that mean we need to know about cancer and life expectancy ? Not really, most of us absorb all sorts of things from reading the newspapers which enable us to make educated guesses.

So lets start with some common sense reasoning. You probably know that smoking kills largely through cancer and heart disease and that these are diseases that arise mainly in the over-fifties. You also know that most people don’t live beyond eighty. So the answer must be somewhere between 0 and 30. It would be 0 if no smoker’s life was ever shortened by the habit, and 30 if every smoker died on his or her 50th birthday. Now clearly it’s not 0, otherwise there wouldn’t be a problem and we wouldn’t have asked the question. So we can say that the answer is at least 1 year and could be as much as 30 years.

## The Simple Rule

What we’ve just established is what mathematicians call an upper and a lower bound , 30 and 1 . Now the simple rule we mentioned above comes into play: you have to work out what is called the ‘geometric mean’ of your upper and lower bounds. You get this by multiplying the upper and lower bounds together and taking the square root. 1 x 30 is 30, and the square root of 30 is just over 5 (5 x 5 is 25, 6 x 6 is 36).

Geometric mean = $\sqrt{( Upper Bound * Lower Bound )}$

With these simple pieces of knowledge – or even just guesswork – we have arrived at some kind of answer – 5, The actual answer is 6.5 so we’ve not done too badly.

## Why does it work ?

simply put … you get an answer that is within the right ball-park.

Getting an answer as close to the correct one as 5 to 6.5 is not a fluke. Using these techniques will usually produce a number with what scientists call the right order of magnitude. This means that your answer will probably be out by no more than a factor of ten. This may not seem very useful and in real life it isn’t. But sometimes in science any answer is better than no answer at all.

You can use the same rough-and-ready techniques to answer questions like ‘How many people are airborne over the USA at any given moment?’ or ‘How long a hot dog can be made from a typical cow?’ and get an answer that is within the right ball-park.