Fermi problem

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A Fermi problem or Fermi question is a quantitative estimate for a problem for which practically no data is initially available. It is named after the nuclear physicist Enrico Fermi , who was known for being able to spontaneously provide good estimates despite a lack of information - for example, during the first atomic bomb test ( Trinity test ) , he threw scraps of paper into the air and observed how far they were blown away by the pressure wave ; from this, he was able to estimate the approximate explosive power of the bomb on site long before the sensor measurements were evaluated.

method

The challenge of such problems is that you neither have direct experience from a similar problem, nor are the necessary data available with which you could directly make a calculation. On the other hand, one knows the connections in the environment of the problem very well and can use them to come to a solution indirectly.

The prerequisite for solving a Fermi problem is therefore a certain general knowledge and “ common sense ”. However, since this prior knowledge cannot be used directly for the solution, this prior knowledge must be quantified and the respective assumptions justified. The overall result can then be determined from these partial estimates - often only in several stages. The lack of empirical values ​​for the overall problem is compensated for by the fact that empirical values ​​are available for partial problems; and the lack of data for the calculation is compensated for from these estimates for the sub-problems.

The overall result is often surprisingly accurate (at least on the right scale). Since the sub-problems are known very well (or could be broken down further), their estimates are quite good and move around the actual values. In addition, there are consistently no systematic errors, rather it is likely that the estimation errors partially cancel each other out - if one variable was estimated as too large, another was perhaps estimated as too small.

Example: piano tuner in Chicago

The classic example of a Fermi problem is the question of the number of piano tuners in Chicago . Initially, there is neither statistical data with which to start a calculation (such as the average number of piano tuners per 1000 inhabitants) nor empirical values ​​from other cities that can be extrapolated to Chicago. However, one knows very well how a piano tuner works; this results in z. B. the following calculation.

Assumptions:

  • About 3 million people live in Chicago.
  • Around two people live in one household on average .
  • Around every twentieth household has a piano that is regularly tuned.
  • Pianos are tuned about once a year.
  • It takes about two hours to tune a piano, including travel time.
  • A piano tuner works 8 hours a day, a 5 day week and works 40 weeks a year.

This gives the number of pianos to be tuned each year in Chicago:

(3,000,000 inhabitants) / (2 people per household) × (1 piano / 20 households) × (1 voices per piano and year) = A piano has to be tuned 75,000 times per year in Chicago.

A piano tuner can do the following work:

(40 weeks per year) × (5 days per week) × (8 hours per day) / (2 hours per piano) = a piano tuner can tune 800 pianos per year.

So there should be around 100 piano tuners in Chicago.

You can use this knowledge in the following way, for example: If you want to open a shop that sells tools for piano tuners, and know from a further estimate that you need 10,000 potential customers, you see that such a shop is even in a big city like Chicago Far from worth it. But you can modify your plans and recalculate each time until you come across a presumably functioning business model .

application

  • Replace calculations: Often you are not interested in the exact result of a calculation, but want to assess the magnitude first . With a good estimate, you can sometimes do without a complex calculation. Or it may be that an exact calculation is not possible because no data is available - an example of this is the Drake equation , which estimates the number of intelligent civilizations in the Milky Way (see also Fermi Paradox ).
  • Prepare calculations: Instead of performing a mathematical “blind flight”, one already knows the order of magnitude through an estimate and can thus assess the effort that has to be made in order to obtain the necessary accuracy . For example, one could estimate the magnitude of a speed and thus assess whether relativistic effects must be taken into account in the precise calculation or whether these can be neglected.
  • Check calculations: A calculation error in a complicated calculation is not noticed so quickly; The result can be checked with a good estimate, because errors are less likely there due to its simplicity, and the estimate also provides intermediate results . (It makes sense to carry out the estimation first, because otherwise you can easily be influenced by the already calculated result when making an estimation.)
  • By breaking it down into sub-problems with many assumptions, the estimate can be improved more easily by striving for more exact values ​​for the individual partial estimates. On the other hand, if you only start from a few more complicated basic assumptions , it is usually more difficult to improve them.
  • By breaking it down into sub-problems, it is usually possible to specify a lower and upper limit for the overall result with little effort by inserting the smallest or largest probable value for each sub-problem.
  • The approach to Fermi problems is fundamental for the natural sciences - by solving Fermi problems one learns to break down a complex problem into simpler sub-problems and to clearly state which theories / assumptions / assumptions are based on. While in science one usually only works on small partial aspects, Fermi problems are usually much broader - they require (due to a lack of prior knowledge) the complete chain of arguments, starting with the description of the basic assumptions; In the scientific community, on the other hand, one takes complete theories for granted and simply builds on them.

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