Friday, September 25, 2009

FPOTW The typical Fermi Problem

This weeks "Fermi Problem of the Week" is taken from a previous blog of a similar name. Here the reader is introduced to the concept of the Fermi problem.
Below is a picture of the Fermi problems names sake, Enrico Fermi.

This image is available from the Archival Research Catalog of the National Archives and Records Administration under the ARC Identifier 558578
Linked from Wikipedia

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, meaning it is accurate to a power of 10, or at best a factor of 2 or 3.

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

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