Thursday, December 29, 2005

The typical Fermi Problem

Fermi problems, or back of the envelope calculations, are a method of estimating an answer to a question where there is seemingly not enough information to answer the question.

The prototype Fermi Problem is as follows:

How many piano tuners are in the city of New York?

One might think there is not enough information to answer this question.
However, by using a bit of common sense and a few reasonable guesses, one can estimate the answer.
For instance, one could go about tackling this question as follows:

First, how many people are in NYC?
We can estimate this as about 8 million.

How many people have a piano? 1 in 10? 1 in 20? Lets use 1 in 50.

That gives us the number of pianos in NYC as 160000.

Lets suppose a piano needs tuned once every 4 years. This means that each year about 40000 pianos need tuned.
Suppose an average piano tuner can tune 4 pianos a day, and works 200 days a year, this means that the average piano tuner tunes 800 pianos a year.

Therefore there are about 40000/800 = 50 piano tuners in NYC.

This seems to be a reasonable number, as 5 would be far to low and 500 would be too many.

A Fermi problem solution is also known as an order of magnitude answer, meening it is acurate to a power of 10, or at best a factor of 2 or 3.

Try this problem or your home town, then afterwards check the phone book. You will find that your answer will be remarkably close.