What is a Fermi problem/ estimate?

During the recent decades of Big Data, machine learning and deep learning together with powerful computing hardware, it is becoming a common practice to deal with voluminous data to seek solutions. As the 1991 economics Nobel laureate Ronald H Coase famously said “ if you torture data enough, it will confess”.

During the Covid Pandemic, there was a frequent reference to enough data not being available to take relevant policy decisions. The situation reminded me of the brilliant Physicist Enrico Fermi after whom a method is named of making quick approximate estimates that can be made without having the full data.

Enrico Fermi (1901-1954) was an Italian physicist who made significant discoveries in nuclear physics and quantum mechanics. In 1938, he received the Nobel Prize in physics for his discovery of nuclear reactions caused by slow neutrons. This mechanism led directly to the development of atomic bombs and nuclear fission reactors. After receiving his Nobel Prize, he migrated with his family to the United States to escape the fascist regime of Benito Mussolini, where he soon began contributing to the Manhattan Project.

Fermi was famous for being able to make good estimates in situations where very little information was available. When the first nuclear bomb was tested, Fermi was nearby to observe. To get a preliminary estimate of the amount of energy released, he sprinkled small pieces of paper in the air and observed what happened when the shock wave reached them. Being so close to the bomb on this and many other occasions exposed Fermi to dangerous radiation that led to his untimely death by cancer at the age of 53. Fermi was aware of the danger, but chose to work on this project anyway because he believed that the work was vital in the fight against Fascism.

Fermi often amused his friends and students by inventing and solving whimsical questions such as “How many piano tuners are there in Chicago?”.

A “Fermi Question” asks for a quick estimate of a quantity that seems difficult or impossible to determine. Fermi’s approach to such questions was to use common sense and rough estimates of quantities to piece together a ball-park value.

For example, one way to estimate the number of piano tuners in Chicago is to break the process into steps: estimate the population; estimate the number of households in the population; estimate the fraction of households that have pianos; estimate how often each household has its piano tuned; estimate the time it takes to tune a piano; estimate how many hours a piano tuner would work each week.

In this case, it is possible to check the estimate by looking in the phone book to see how many piano tuners are actually in Chicago.

Fermi was famous for being able to make good estimates in situations where very little information was known.

In the New Education Policy 2020, several references have been made to critical thinking. One way of encouraging this at School level may be the setting up of Fermi Questions Lab. A pool of questions can be created that is appropriate to the level of the learner. But instead of gradation by each year, it could be according to the stages proposed in the New education policy.

So one set for classes 9 to 12 and another for classes 6 to 8. Younger children may not be able to appreciate and enjoy dealing with such problems.

During the lab sessions here, students would choose questions from a pool of “ Fermi questions”. This activity may also be done in small teams to encourage collaboration and teamwork. The format of the submission of the report on each question could be :

1. Question: State the question and discuss how you will interpret it.

2. Wild Guess: What is your answer without any calculating?

3. Educated Guess: List the pieces of information you will need to answer this Fermi question more precisely. Estimate the value of each quantity in your list. Based on your estimates, what is your solution to the Fermi question? Show all your steps and use words to explain them.

4. Variables and Formulas: Choose variable names for each quantity that you estimated. Write a series of formulas or a procedure that explains how you used the quantities to find the solution. Try to simplify the process into a single formula that answers the Fermi question if possible.

5. Gathering Data: Perform experiments, conduct surveys, make measurements, or search for information that would help you to obtain a more precise estimate.

6. Conclusions: State your final answers to the question. Explain some possible sources of error in your procedure.

Fermi questions encourage creative thinking involving different solution strategies so they promote a range of problem-solving skills requiring students to be logical and inventive. Students would like Fermi questions because they are:

- Open-ended problems
- Have no exact answer, no definite solution
- Interesting and motivating questions
- Challenging and rewarding
- Create a culture of questioning and curiosity

Fermi questions help develop much-need estimation skills and support the ‘feel’ of whether an answer is reasonable or not.

The Drake equation:

The same set f ideas that Fermi used to estimate the number of piano tuners in Chicago can be extended to estimate the number of communicating civilizations in the cosmos, or more simply put, the odds of finding intelligent life in the universe.

First proposed by radio astronomer Frank Drake in 1961, the equation calculates the number of communicating civilizations by multiplying several variables. It’s usually written, according to the Search for Extraterrestrial Intelligence (SETI), as:

**N** = The number of civilizations in the Milky Way galaxy whose electromagnetic emissions are detectable.

**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.

The challenge (at least for now) is that astronomers don’t have firm numbers on any of those variables, so any calculation of the Drake Equation remains a rough estimate for now. There have been, however, discoveries in some of these fields that give astronomers a better chance of finding the answer. More information on the Drake equation : https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Supplemental_Modules_(Astronomy_and_Cosmology)/Astronomy/Life_beyond_the_Earth/The_Drake_Equation

Nice article