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Grappling with Fermi Problems

Team Latte
18th March, 2014

[Compiled and modified, with permission, from FEM lectures by Rahul Bhattacharya between 2009 and 2011]

[Special Note: As we compile this article, our hearts reach out to all those who had their loved ones on board the Malaysian Airlines flight MH370 en route to Beijing from Kuala Lumpur on the morning of Saturday, 8th March, 2014 and all passengers on board that flight are in our thoughts and prayers.]

Say, you needed an answer this question for whatever reason: how many derivatives traders are there in the Greater New York area who earned more than $1 million in bonus in 2008? In the absence of any other information or data this may be an extremely difficult question to answer. Or, consider another question: how many derivatives traders have read Nassim Taleb's book Dynamic Hedging? Once again this is, apparently, a very difficult, if not an, impossible question to answer. But believe it or not answers to both these questions can be found, albeit with great approximations.

Take, for example, the recent tragedy of Malaysian Airlines flight MH370. With regard that let's pose another seemingly impossible question to answer:

Given the fact that the Malaysian Airlines flight MH370 left Kuala Lumpur on Saturday, 8th March, 2014 at 12:41 a.m. and given the fact that the plane got deliberately diverted off its scheduled airway and given the fact that at around 1:19 a.m., 12 minutes after the plane had changed course to the west, co-pilot Fariq Abdul Hamid made a routine sign off "All right, good night" as his final radio call to the Malaysian air traffic controllers and, finally, given the fact that no real debris of the aircraft has yet been discovered during an extensive and almost exhaustive search operation scouring immensely vast areas of land and ocean where is the aircraft (or, the crash debris of the aircraft) now?

To most of us, finding the correct (exact) - or, even a very good - answer to the above question may seem very challenging, at the very least. Say, we didn't have any satellite data on the location of the debris and no remote "handshake" pings were detected by the satellites after the final voice sign off by the co-pilot and transponder black-out. Then, it would be almost impossible to locate the missing aircraft. Yet, in the absence of any relevant and helpful data, using very broad approximations, including those regarding airspeed, altitude when the aircraft made last voice contact, the last transponder signal, etc., one can still locate the coordinates of the missing (crashed) aircraft with certain degree of accuracy.

All the above questions (problems) are very similar to what is known in mathematical modeling as a Fermi problem (or a Fermi question). It is named after the Nobel Prize winning Italian physicist Enrico Fermi, best known for his pioneering work in many areas of nuclear physics, including the development of the atomic bomb as part of the Manhattan project.

Fermi problems (questions) are seemingly impossible problems to solve and yet if one applies broad approximations and makes rough estimates of the input parameters needed to solve the problem, an approximate solution can be arrived at.

Fermi, who was as great a teacher as he was a researcher, had tremendous knack for posing very difficult questions to his students, the answers to which almost seemed impossible to guess. Some of his questions were, "How many railroad cars are there in the U.S.?" or "How many piano tuners are there in Chicago?" Prima facie, such problems suffer from the lack of data, something what is now happening with the Malaysian Airlines problem. Either the data is completely missing or highly insufficient. Yet, when you work around the problem by making reasonably educated guess - approximations - around the limited data that is available you end up finding an approximate solution to the problem. And if the approximations are to the nearest power of 10, then that answer is fine. Physicists refer to a factor of 10 error as being within cosmological accuracy.

Fermi problems involve estimating physical quantities by using intelligent approximation. Fermi was of the opinion that any "thinking person" should be able to estimate any quantity - such as, how many licensed carpenters are there in the city of London - with great degree of accuracy simply by using "one's own head". Within the context of problems in physics Fermi problems should end up in estimates to the nearest power of ten simply by using logical analysis, i.e. without using reference books, calculators or computers.

The mathematics behind Fermi problems as follows: we usually arrive at the "quantity" (answer to a problem) that we want to determine by multiplying together (or dividing by) many other quantities whose values we can easily estimate. There will be some uncertainty associated with each of these. There will be overestimates and underestimates. However, we consider the product of these quantities; the uncertainty will combine in some way or cancel out. Remember that when we multiply quantities, we actually add their logarithms. While working with these uncertainties, we end with a lognormal distribution because if a large number of independent random variables are multiplied, the result converges on a log-normal probability distribution.

[DISCLAIMER: Facts, data and information regarding the flight MH370 are taken from various media/online stories on the internet and Risk Latte Company ("the Company") and/or any other member, staff or associate of the Company, including but not limited to the author of this article, can vouch for the accuracy or validity of the same. Data and factual information used for illustration and exposition only.]

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