The art of guessing -Fermi Estimations

“ How many times could you say the alphabet in 24 hours?”

“ How many hairs are present on your head?”

How would one even approach questions like this?

Fermi Estimation is the answer.

Enrico Fermi was an Italian American physicist and the creator of the world’s first nuclear reactor, the Chicago Pile-1. He has been called “the architect of the nuclear age”.

Fermi was known for his ability to make good approximate calculations with little or no actual data. He was able to estimate the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi’s estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons.

So how did he do this?

Well, the Trinity test approach requires knowledge of physics and so I will leave a link for the explanation for interested folks here.

How did Fermi estimate the power of the Trinity bomb?

The entire process revolves around dividing the main problem into various subproblems and solving them. Let's look at the classic Fermi question.

The classic Fermi problem, generally attributed to Fermi, is “How many piano tuners are there in Chicago?” A typical solution to this problem would involve multiplying together a series of estimates that would yield the correct answer if the estimates were correct. For example, we might make the following assumptions:

1. There are approximately 5,000,000 people living in Chicago.
2. On average, there are two persons in each household in Chicago.
3. Roughly one household in twenty has a piano that is tuned regularly.
4. Pianos that are tuned regularly are tuned on average about once per year.
5. It takes a piano tuner about two hours to tune a piano, including travel time.
6. Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year.

From these assumptions we can compute that the number of piano tunings in a single year in Chicago is:

(5,000,000 persons in Chicago) / (2 persons/household) × (1 piano/20 household) × (1 piano tuning per piano per year) = 125,000 piano tunings per year in Chicago.

And we can similarly calculate that the average piano tuner performs: (50 weeks/year)×(5 days/week)×(8 hours/day)×(1 piano tuning per 2 hours per piano tuner) = 1000 piano tunings per year per piano tuner.

Dividing gives (125,000 piano tuning per year in Chicago) / (1000 piano tunings per year per piano tuner) = 125 piano tuners in Chicago.

Let's see another one.

How many hairs are present on your head?

Well, the initial reaction of anyone to this question would be that it is absurd but we just have to make a guess. (A calculated one).

Let's assume that we have 150 hairs per square cm on our head. Or you can wax an area of 1cm² on your scalp and count the number of hairs that are removed.

Assume the head to be spherical and say around 50% of the sphere is covered with hair. Assuming the radius to be around 10 cm we can say that the area covered by hairs = 1/2(4∗pi∗r²)=200∗pi = 628 cm²

So the amount of hair on a human head = 628 * 150 = 94,200 hairs.

This value is actually very close to the average amount of hairs in the human head which is about 100000 hairs.

Here are more examples and explanations regarding Fermi Estimation by Brilliant.

Fermi Estimate | Brilliant Math & Science Wiki

But why should you learn or implement this technique at all?

Fermi estimation has applications in any field that would require making calculated assumptions.

For Software Engineers —

How much data does Youtube store?

For startups —

Fermi estimation for startup business models

For Data Science —

Fermi Problem Solving for Data Scientists - Data Scientist Insights