--- title: "Fermi Calculation Examples" description: "Fermi estimates or problems are quick heuristic solutions to apparently insoluble quantitative problems rewarding clever use of real-world knowledge and critical thinking; bibliography of some examples." created: 2019-03-29 modified: 2019-03-29 status: in progress confidence: possible importance: 4 css-extension: dropcaps-yinit ...
> A short discussion of "Fermi calculations": quick-and-dirty approximate answers to quantitative questions which prize cleverness in exploiting implications of common knowledge or basic principles in given reasonable answers to apparently unanswerable questions. > > Links to discussions of Fermi estimates, and a list of some Fermi estimates I've done.
I enjoy [Fermi problems](!W) ([LessWrong](https://www.lesswrong.com/posts/PsEppdvgRisz5xAHG/fermi-estimates))---it's like [dimensional analysis](!W) for everything outside of physics^[This is a little misleading; dimensional analysis is much more like [type-checking](!W) a program in a language with a good type system like [Haskell](https://en.wikipedia.org/wiki/Haskell). Given certain data types as *inputs* and certain allowed transformations on those data types, what data types *must* be the resulting output? But the analogy is still useful.]. Not only are they fun to think about, they can be amazingly accurate, and are extremely cheap to do---because they are so easy, you do them in all sorts of situations you wouldn't do a 'real' estimate for, and are a fun part of a [physics education](https://arxiv.org/abs/1109.1165 "'Fermi Problem: Power developed at the eruption of the Puyehue-Cordón Caulle volcanic system in June 2011', Asorey & Dávalos 2011"). The common distaste for them baffles me; even if you never work through Hubbard's [_How to Measure Anything_](https://www.amazon.com/How-Measure-Anything-Intangibles-Business/dp/1118539273/) ([some strategies](https://80000hours.org/2013/05/estimation-part-i-how-to-do-it/ "Estimation—Part I: How to do it?")) or [_Street-Fighting Mathematics_](https://www.amazon.com/Street-Fighting-Mathematics-Educated-Guessing-Opportunistic/dp/026251429X) or read [Douglas Hofstadter's](!W "Douglas Hofstadter") 1982 essay ["On Number Numbness"](/doc/math/1982-hofstadter-2.pdf) (collected in _[Metamagical Themas](https://en.wikipedia.org/wiki/Metamagical_Themas "‘Metamagical Themas: Questing for the Essence of Mind and Pattern’, Hofstadter 1982")_), it's something you can teach yourself by asking, what information is public available, what can I compare this too, how can I put various boundaries around the true answers^[eg. if someone asks you how many piano tuners there are in Chicago, don't look blank, start thinking! 'Well, there must be fewer than 7 billion, because the human race isn't made of piano tuners, and likewise fewer than 300 million (the population of the United States), and heck, Wikipedia says Chicago has only 2.6 million people and piano tuners are rare, so there must be many fewer than *that*...' You always know *something*, and have a universe of beliefs which imply constraints on everything and equilibria.] An entire ["estimation" subreddit](https://www.reddit.com/r/estimation/) is devoted to working through questions like these (it can be quite fun), and of course, there are the memorable ["what if?"](https://what-if.xkcd.com/) _Xkcd_ columns. [Timothy Gowers](!W) suggests [a number of problems](https://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/ "How should mathematics be taught to non-mathematicians?") which might help children really learn how to think with & apply the math they learn. To look further afield, here's a quick and nifty application by investor John Hempton to the [Sino Forestry fraud](!W "Sino-Forest Corporation#Fraud allegations"): ["Risk management and sounding crazy"](https://brontecapital.blogspot.com/2011/09/risk-management-and-sounding-crazy.html). What I found most interesting about this post was not the overall theme that the whistleblowers were discounted before and after they were proven right (we see this in many bubbles, for example, the housing bubble), but how one could use a sort of [Outside View](https://www.lesswrong.com/w/inside-outside-view)/Fermi calculation to sanity-check the claims. If Sino Forestry was really causing 17m cubic meters of wood to be processed a year, where was all the processing? This simple question tells us a *lot*. With medicine, there is one simple question one can always ask too---where is the increased longevity/all-cause mortality reduction? (This is an important question to ask of studies, such as a [recent caloric restriction study](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2812811/ "‘Caloric restriction delays disease onset and mortality in rhesus monkeys’, Colman et al 2009").) Simple questions and reasoning can tell us a lot. # My Fermi Examples You especially want to do Fermi calculations in areas where the data is unavailable; I wind up pondering such areas frequently: - [is a lip-reading website a good idea?](/startup-idea#lipreading-website "'Startup Ideas § Lipreading Website', Gwern 2017"){#gwern-startup-ideas} - how many women [dye their hair blonde](/note/note#somatic-genetic-engineering) - how many [people does Folding@home kill](/charity-is-not-about-helping "'Charity is not about Helping', Gwern 2011") - what's [the entropy of natural language](/note/note#efficient-natural-language) - or [how big a computer](/simulation-inference "'Simulation Inferences', Gwern 2009") is needed to compute the universe - do [men have shorter *real* lives than women](/note/note#who-lives-longer-men-or-women) - how much do Girl Scouts cookies [cost and earn them](/girl-scouts#cookie-prices-and-inflation) - how many people have [used modafinil](/modafinil#side-effects), and what is the cheapest we can expect to find [modafinil for](/modafinil#margin-estimation) - quickly assessing the rough probability of some event as I made one of my [thousands of predictions](/prediction-market "'Prediction Markets', Gwern 2009") - how many sellers are there on [Silk Road 1](/silk-road "'Silk Road 1: Theory & Practice', Gwern 2011") - checking whether posthumous organ donation [justifies the Chinese justice system](https://news.ycombinator.com/item?id=3389084) (no) - [how many tourists](https://www.lesswrong.com/posts/etAJHCL7GwPFTFRAv/smbc-comic-poorly-programmed-average-utility-maximizing-ai?commentId=nwH8WbPNG63ZtY2hS) to the Egyptian pyramids there have been compared to the workers who build them - [what we can infer from adultery & false paternity rates](/note/note#politicians-are-not-unethical) - [how many times have video gamers killed Mario](https://www.reddit.com/r/estimation/comments/105okb/request_how_many_times_has_mario_died/) - is tiger skin [more profitable](https://www.reddit.com/r/WTF/comments/188zui/this_is_beyond_saddening_tigers_starved_to_make/c8cql4x/) than tiger bone wine? - [plating Versailles with gold](https://slatestarcodex.com/2013/03/03/reactionary-philosophy-in-an-enormous-planet-sized-nutshell/#comment-544) - the [number of people killed by falling pianos](https://www.lesswrong.com/posts/PsEppdvgRisz5xAHG/fermi-estimates8pts) - [the plausibility of](https://www.reddit.com/r/anime/comments/1cqtae/in_honor_of_kuronekos_birthday_my_favorite_scene/c9ja9a5/ "‘Plausibility of Oreimo’s Kirino’, Gwern 2013") [Oreimo's](!W "Oreimo") Kirino Kosaka character - [how many integers one to a million do you see over a lifetime?](https://www.reddit.com/r/estimation/comments/1f765d/request_in_an_average_lifetime_how_many_of_the/ca85zcl "‘How many numbers does one see per minute?’, Gwern 2013") (R simulation + [Benford’s law](!W)) - [how many anecdotes of CrossFit causing rhabdomyolysis does it take to justify "CrossFit causes rhabdomyolysis"?](https://www.lesswrong.com/posts/aARGW967NexuSrCwW/rationality-competitiveness-and-akrasia?commentId=ggmadikyQcspwpfA3 "‘CrossFit rhabdo rates’, Gwern 2013") - [how long would it take to read every book in existence?](https://www.reddit.com/r/estimation/comments/27i4wq/request_how_long_would_it_take_to_read_every_book/ci1jfno/ "‘How long would it take to read every book ever written if every human started reading now?’, Gwern 2014") - [what would it cost to replace the US nuclear arsenal?](https://www.reddit.com/r/estimation/comments/2chtft/if_all_the_uss_atomic_bombs_disappeared_assuming/cjfyses/ "‘It would cost >$100b to replace all the warheads in the US nuclear arsenal’, Gwern 2014") - [What are the lifetime odds of being pooped on by a bird?](https://www.reddit.com/r/estimation/comments/4f9bxl/what_are_the_lifetime_odds_of_being_pooped_on_by/ "‘The Warm Equation of the lifetime odds of being pooped on by a bird’, Gwern 2016") - **See Also**: [Fake Journal Club: Teaching Critical Reading](/fake-journal-club){.backlink-not}, [Little's law](!W); [Frink](https://frinklang.org/)