Fermi problem
Encyclopedia
In science
, particularly in physics
or engineering
education
, a Fermi problem, Fermi question, or Fermi estimate is an estimation
problem designed to teach dimensional analysis
, approximation
, and the importance of clearly identifying one's assumptions. Named after physicist Enrico Fermi
, such problems typically involve making justified guesses about quantities that seem impossible to compute given limited available information.
Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of 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, a difference of less than one order of magnitude
.
?" 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:
From these assumptions we can compute that the number of piano tunings in a single year in Chicago is
/ (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 125,000 piano tunings per year in Chicago.
We can similarly calculate that the average piano tuner performs
×(5 days/week)×(8 hours/day)/(2 hours to tune a piano) = 1000 piano tunings per year per piano tuner.
Dividing gives
/ (1000 piano tunings per year per piano tuner) = 125 piano tuners in Chicago.
A famous example of a Fermi-problem-like estimate is the Drake equation
, which seeks to estimate the number of intelligent civilizations in the galaxy. The basic question of why, if there is a significant number of such civilizations, ours has never encountered any others is called the Fermi paradox
.
Fermi estimates are also useful in approaching problems where the optimal choice of calculation method depends on the expected size of the answer. For instance, a Fermi estimate might indicate whether the internal stresses of a structure are low enough that it can be accurately described by linear elasticity
; or if the estimate already bears significant relationship in scale
relative to some other value, for example, if a structure will be over-engineered to withstand loads several times greater than the estimate.
Although Fermi calculations are often not accurate, as there may be many problems with their assumptions, this sort of analysis does tell us what to look for to get a better answer. For the above example, we might try to find a better estimate of the number of pianos tuned by a piano tuner in a typical day, or look up an accurate number for the population of Chicago. It also gives us a rough estimate that may be good enough for some purposes: if we want to start a store in Chicago that sells piano tuning equipment, and we calculate that we need 10,000 potential customers to stay in business, we can reasonably assume that the above estimate is far enough below 10,000 that we should consider a different business plan (and, with a little more work, we could compute a rough upper bound on the number of piano tuners by considering the most extreme reasonable values that could appear in each of our assumptions).
A Fermi calculation that involves the multiplication of several estimated factors (e.g. the number of piano tuners in Chicago) will probably be more accurate than might be first supposed (assuming that there is no consistent bias in the estimated factors). This is because if there is no consistent bias, then there will probably (with a binomial distribution) be some factors that are estimated too high and other factors that are estimated too low, and such errors will partially cancel each other out.
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
, particularly in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
or engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
education
Education
Education in its broadest, general sense is the means through which the aims and habits of a group of people lives on from one generation to the next. Generally, it occurs through any experience that has a formative effect on the way one thinks, feels, or acts...
, a Fermi problem, Fermi question, or Fermi estimate is an estimation
Estimation
Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...
problem designed to teach dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
, approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
, and the importance of clearly identifying one's assumptions. Named after physicist Enrico Fermi
Enrico Fermi
Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
, such problems typically involve making justified guesses about quantities that seem impossible to compute given limited available information.
Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test
Trinity test
Trinity was the code name of the first test of a nuclear weapon. This test was conducted by the United States Army on July 16, 1945, in the Jornada del Muerto desert about 35 miles southeast of Socorro, New Mexico, at the new White Sands Proving Ground, which incorporated the Alamogordo Bombing...
, 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, a difference of less than one order of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...
.
Examples of Fermi problems
The classic Fermi problem, generally attributed to Fermi, is "How many piano tuners are there in ChicagoChicago
Chicago is the largest city in the US state of Illinois. With nearly 2.7 million residents, it is the most populous city in the Midwestern United States and the third most populous in the US, after New York City and Los Angeles...
?" 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:
- There are approximately 5,000,000 people living in Chicago.
- On average, there are two persons in each household in Chicago.
- Roughly one household in twenty has a piano that is tuned regularly.
- Pianos that are tuned regularly are tuned on average about once per year.
- It takes a piano tuner about two hours to tune a piano, including travel time.
- 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
/ (2 persons/household) × (1 piano/20 households) × (1 piano tuning per piano per year) = 125,000 piano tunings per year in Chicago.
We can similarly calculate that the average piano tuner performs
×(5 days/week)×(8 hours/day)/(2 hours to tune a piano) = 1000 piano tunings per year per piano tuner.
Dividing gives
/ (1000 piano tunings per year per piano tuner) = 125 piano tuners in Chicago.
A famous example of a Fermi-problem-like estimate is the Drake equation
Drake equation
The Drake equation is an equation used to estimate the number of detectable extraterrestrial civilizations in the Milky Way galaxy. It is used in the fields of exobiology and the Search for ExtraTerrestrial Intelligence...
, which seeks to estimate the number of intelligent civilizations in the galaxy. The basic question of why, if there is a significant number of such civilizations, ours has never encountered any others is called the Fermi paradox
Fermi paradox
The Fermi paradox is the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilizations and the lack of evidence for, or contact with, such civilizations....
.
Advantages and scope
Scientists often look for Fermi estimates of the answer to a problem before turning to more sophisticated methods to calculate a precise answer. This provides a useful check on the results: where the complexity of a precise calculation might obscure a large error, the simplicity of Fermi calculations makes them far less susceptible to such mistakes. (Performing the Fermi calculation first is preferable because the intermediate estimates might otherwise be biased by knowledge of the calculated answer.)Fermi estimates are also useful in approaching problems where the optimal choice of calculation method depends on the expected size of the answer. For instance, a Fermi estimate might indicate whether the internal stresses of a structure are low enough that it can be accurately described by linear elasticity
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
; or if the estimate already bears significant relationship in scale
Scale
-Length:* Architect's scale, a ruler-like device which facilitates the production of technical drawings* Engineer's scale, a ruler-like device similar to the Architect's scale, they are helpful when drawing rooms...
relative to some other value, for example, if a structure will be over-engineered to withstand loads several times greater than the estimate.
Although Fermi calculations are often not accurate, as there may be many problems with their assumptions, this sort of analysis does tell us what to look for to get a better answer. For the above example, we might try to find a better estimate of the number of pianos tuned by a piano tuner in a typical day, or look up an accurate number for the population of Chicago. It also gives us a rough estimate that may be good enough for some purposes: if we want to start a store in Chicago that sells piano tuning equipment, and we calculate that we need 10,000 potential customers to stay in business, we can reasonably assume that the above estimate is far enough below 10,000 that we should consider a different business plan (and, with a little more work, we could compute a rough upper bound on the number of piano tuners by considering the most extreme reasonable values that could appear in each of our assumptions).
A Fermi calculation that involves the multiplication of several estimated factors (e.g. the number of piano tuners in Chicago) will probably be more accurate than might be first supposed (assuming that there is no consistent bias in the estimated factors). This is because if there is no consistent bias, then there will probably (with a binomial distribution) be some factors that are estimated too high and other factors that are estimated too low, and such errors will partially cancel each other out.
See also
- HandwavingHandwavingHandwaving is a pejorative label applied to the action of displaying the appearance of doing something, when actually doing little, or nothing. For example, it is applied to debate techniques that involve logical fallacies. It is also used in working situations where productive work is expected,...
- Orders of approximationOrders of approximationIn science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...
- Stein's exampleStein's exampleStein's example , in decision theory and estimation theory, is the phenomenon that when three or more parameters are estimated simultaneously, there exist combined estimators more accurate on average than any method that handles the parameters separately...
- Back-of-the-envelope calculationBack-of-the-envelope calculationA back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as the actual back of an envelope...
- HeuristicHeuristicHeuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
Further reading
There are a number of university level courses devoted to estimation and the solution of Fermi problems. The materials for these courses are a good source for additional Fermi problem examples and material about solution strategies- 6.055J / 2.038J The Art of Approximation in Science and Engineering taught by Sanjoy Mahajan at the Massachusetts Institute of TechnologyMassachusetts Institute of TechnologyThe Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
(MIT). - Physics on the Back of an Envelope taught by Lawrence Weinstein at Old Dominion UniversityOld Dominion UniversityOld Dominion University is a state university located in Norfolk, Virginia, United States, and is accredited by the Southern Association of Colleges and Schools...
. - Order of Magnitude Physics taught by Sterl Phinney and Peter Goldreich at the California Institute of TechnologyCalifornia Institute of TechnologyThe California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
. - Chapter 2: Discoveries on the Back of an Envelope from Frontiers of Science: Scientific Habits of Mind taught by David HelfandDavid HelfandDavid J. Helfand is chair of the Department of Astronomy at Columbia University as well as the co-director of the Columbia Astrophysics Laboratory. He has also served as part of the university's Physics Department. His stated research interests include radio surveys, the origin and evolution of...
at Columbia UniversityColumbia UniversityColumbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
External links
- The University of MarylandUniversity of Maryland, College ParkThe University of Maryland, College Park is a top-ranked public research university located in the city of College Park in Prince George's County, Maryland, just outside Washington, D.C...
Physics Education Group maintains a collection of Fermi problems. - An example of a Fermi Problem relating to total gasoline consumed by cars since the invention of cars- and compares it to the output of the energy released by the sun.
- Lawrence Weinstein & John A. Adam, Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin Princeton University Press. 2008. ISBN 0691129495. ISBN 978-1-4008-2444-1. A textbook on Fermi problems.
- Aaron Santos, How Many Licks?: Or, How to Estimate Damn Near Anything. Running Press. 2009. ISBN 0762435607. ISBN 978-0762435609.