Fermi Estimates

Correction:

(2 Fermis per day)×(90 days) = 200 Fermi estimates.

1Smath3y

2 fermis a day x 3000 days = 6000 fermi estimates

It's probably worth figuring out what went wrong in Approach 1 to Example 1, which I think is this part:

[300 cities of 10,000 or more people per county] × [2500 counties in the USA]

Note that this gives 750,000 cities of 10,000 or more people in the US, for a total of at least 7.5 billion people in the US. So it's already clearly wrong here. I'd say 300 cities of 10,000 people or more per county is way too high; I'd put it at more like 1 (Edit: note that this gives at least 250 million people in the US and that's about right). This brings down the final estimate from this approach by a factor of 300, or down to 3 million, which is much closer.

(Verification: I just picked a random US state and a random county in it from Wikipedia and got Bartow County, Georgia, which has a population of 100,000. That means it has at most 10 cities with 10,000 or more people, and going through the list of cities it actually looks like it only has one such city.)

This gives about 2,500 cities in the US total with population 10,000 or more. I can't verify this number, but according to Wikipedia there are about 300 cities in the US with population 100,000 or more. Assuming the populations of cities a... (read more)

[-]elharo13y370

I feel compelled to repeat this old physics classic:

How Fermi could estimate things!
Like the well-known Olympic ten rings,
And the one-hundred states,
And weeks with ten dates,
And birds that all fly with one... wings.

:-)

2VCavallo13y

That is beautiful.

[-]fiddlemath13y270

I've run meetups on this topic twice now. Every time I do, it's difficult to convince people it's a useful skill. More words about when estimation is useful would be nice.

In most exercises that you can find on Fermi calculations, you can also actually find the right answer, written down somewhere online. And, well, being able to quickly find information is probably a more useful skill to practice than estimation; because it works for non-quantified information too. I understand why this is; you want to be able to show that these estimates aren't very far off, and for that you need to be able to find the actual numbers somehow. But that means that your examples don't actually motivate the effort of practicing, they only demonstrate how.

I suspect the following kinds of situations are fruitful for estimation:

Deciding in unfamiliar situations, because you don't know how things will turn out for you. If you're in a really novel situation, you can't even find out how the same decision has worked for other people before, and so you have to guess at expected value using the best information that you can find.
Value of information calculations, like here and here, where you cannot possib

... (read more)

[-]AnnaSalamon13y300

Fermi's seem essential for business to me. Others agree; they're taught in standard MBA programs. For example:

Can our business (or our non-profit) afford to hire an extra person right now? E.g., if they require the same training time before usefulness that others required, will they bring in more revenue in time to make up for the loss of runway?
If it turns out that product X is a success, how much money might it make -- is it enough to justify investigating the market?
Is it cheaper (given the cost of time) to use disposable dishes or to wash the dishes?
Is it better to process payments via paypal or checks, given the fees involved in paypal vs. the delays, hassles, and associated risks of non-payment involved in checks?

And on and on. I use them several times a day for CFAR and they seem essential there.

They're useful also for one's own practical life: commute time vs. rent tradeoffs; visualizing "do I want to have a kid? how would the time and dollar cost actually impact me?", realizing that macademia nuts are actually a cheap food and not an expensive food (once I think "per calorie" and not "per apparent size of the container"), and so on and so on.

[-]fiddlemath13y140

Oh, right! I actually did the comute time vs. rent computation when I moved four months ago! And wound up with a surprising enough number that I thought about it very closely, and decided that number was about right, and changed how I was looking for apartments. How did I forget that?

Thanks!

6A1987dM13y

But calories aren't the only thing you care about -- the ability to satiate you also matters. (Seed oil is even cheaper per calorie.)

[-]Kindly13y220

The main use I put Fermi estimates to is fact-checking: when I see a statistic quoted, I would like to know if it is reasonable (especially if I suspect that it has been misquoted somehow).

[-]lukeprog13y150

Qiaochu adds:

If you get [an estimate] that's obviously wrong, you've discovered an opportunity to fix some aspect of your model of the world.

9Qiaochu_Yuan13y

I also think Fermi calculations are just fun. It makes me feel totally awesome to be able to conjure approximate answers to questions out of thin air.

[-]sketerpot13y250

There's a free book on this sort of thing, under a Creative Commons license, called Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving. Among the fun things in it:

Chapter 1: Using dimensional analysis to quickly pull correct-ish equations out of thin air!

Chapter 2: Focusing on easy cases. It's amazing how many problems become simpler when you set some variables equal to 1, 0, or ∞.

Chapter 3: An awful lot of things look like rectangles if you squint at them hard enough. Rectangles are nice.

Chapter 4: Drawing pictures can help. Humans are good at looking at shapes.

Chapter 5: Approximate arithmetic in which all numbers are either 1, a power of 10, or "a few" -- roughly 3, which is close to the geometric mean of 1 and 10. A few times a few is ten, for small values of "is". Multiply and divide large numbers on your fingers!

... And there's some more stuff, too, and some more chapters, but that'll do for an approximate summary.

[-]Morendil13y210

XKCD's What If? has some examples of Fermi calculations, for instance at the start of working out the effects of "a mole of moles" (similar to a mole of choc donuts, which is what reminded me).

[-]Benquo13y180

Thanks, Luke, this was helpful!

There is a sub-technique that could have helped you get a better answer for the first approach to example 1: perform a sanity check not only on the final value, but on any intermediate value you can think of.

In this example, when you estimated that there are 2500 counties, and that the average county has 300 towns with population greater than 10,000, that implies a lower bound for the total population of the US: assuming that all towns have exactly 10,000 people, that gets you a US population of 2,500x300x10,000=7,500,000,000! That's 7.5 billion people. Of course, in real life, some people live in smaller towns, and some towns have more then 10,000 people, which makes the true implied estimate even larger.

At this point you know that either your estimate for number of counties, or your estimate for number of towns with population above 10,000 per county, or both, must decrease to get an implied population of about 300 million. This would have brought your overall estimate down to within a factor of 10.

[-]tgb13y140

I had the pleasure the other day of trying my hand on a slightly unusual use of Fermi estimates: trying to guess whether something unlikely has ever happened. In particular, the question was "Has anyone ever been killed by a falling piano as in the cartoon trope?" Others nearby at the time objected, "but you don't know anything about this!" which I found amusing because of course I know quite a lot about pianos, things falling, how people can be killed by things falling, etc. so how could I possibly not know anything about pianos falling and killing people? Unfortunately, our estimate gave it at around 1-10 deaths by piano-falling so we weren't able to make a strong conclusion either way over whether this happened. I would be interested to hear if anyone got a significantly different result. (We only considered falling grands or baby grands to count as upright pianos, keyboards, etc. just aren't humorous enough for the cartoon trope.)

[-]gwern13y250

I'll try. Let's see, grands and baby grands date back to something like the 1700s; I'm sure I've heard of Mozart or Beethoven using pianos, so that gives me a time-window of 300 years for falling pianos to kill people in Europe or America.

What were their total population? Well, Europe+America right now is, I think, something like 700m people; I'd guess back in the 1700s, it was more like... 50m feels like a decent guess. How many people in total? A decent approximation to exponential population growth is to simply use the average of 700m and 50m, which is 325, times 300 years, 112500m person-years, and a lifespan of 70 years, so 1607m persons over those 300 years.

How many people have pianos? Visiting families, I rarely see pianos; maybe 1 in 10 had a piano at any point. If families average a size of 4 and 1 in 10 families has a piano, then we convert our total population number to, (1607m / 4) / 10, 40m pianos over that entire period.

But wait, this is for falling pianos, not all pianos; presumably a falling piano must be at least on a second story. If it simply crushes a mover's foot while on the porch, that's not very comedic at all. We want genuine verticality, real free fall. So... (read more)

[-]Qiaochu_Yuan13y300

0 is within an order of magnitude of 3.9, after all.

No it's not! Actually it's infinitely many orders of magnitude away!

9NancyLebovitz13y

Nitpick alert: I believe pianos used to be a lot more common. There was a time when they were a major source of at-home music. On the other hand, the population was much smaller then, so maybe the effects cancel out.

2gwern13y

I wonder. Pianos are still really expensive. They're very bulky, need skilled maintenance and tuning, use special high-tension wires, and so on. Even if technological progress, outsourcing manufacture to China etc haven't reduced the real price of pianos, the world is also much wealthier now and more able to afford buying pianos. Another issue is the growth of the piano as the standard Prestigious Instrument for the college arms races (vastly more of the population goes to college now than in 1900) or signaling high culture or modernity (in the case of East Asia); how many pianos do you suppose there are scattered now across the USA compared to 1800? Or in Japan and China and South Korea compared to 1900? And on the other side, people used to make music at home, yes - but for that there are many cheaper, more portable, more durable alternatives, such as cut-down versions of pianos.

5Richard_Kennaway13y

Concert grands, yes, but who has room for one of those? Try selling an old upright piano when clearing a deceased relative's estate. In the UK, you're more likely to have to pay someone to take it away, and it will just go to a scrapheap. Of course, that's present day, and one reason no-one wants an old piano is that you can get a better electronic one new for a few hundred pounds. But back in Victorian times, as Nancy says elsethread, a piano was a standard feature of a Victorian parlor, and that went further down the social scale that you are imagining, and lasted at least through the first half of the twentieth century. Even better-off working people might have one, though not the factory drudges living in slums. It may have been different in the US though.

4gwern13y

Certainly: http://www.nytimes.com/2012/07/30/arts/music/for-more-pianos-last-note-is-thud-in-the-dump.html?_r=2&ref=arts But like diamonds (I have been told that you cannot resell a diamond for anywhere near what you paid for it), and perhaps for similar reasons, I don't think that matters to the production and sale of new ones. That article supports some of my claims about the glut of modern pianos and falls in price, and hence the claim that there may be unusually many pianos around now than in earlier centuries: At least, if we're comparing against the 1700s/1800s, since the article then goes on to give sales figures: (Queen Victoria died in 1901, so if this golden age 1900-1930 also populated parlors, it would be more accurate to call it an 'Edwardian parlor'.)

2OrphanWilde13y

We got ~$75 for one we picked up out of somebody garbage in a garage sale, and given the high interest we had in it, probably could have gotten twice that. (Had an exchange student living with us who loved playing the piano, and when we saw it, we had to get it - it actually played pretty well, too, only three of the chords needed replacement. It was an experience loading that thing into a pickup truck without any equipment. Used a trash length of garden hose as rope and a -lot- of brute strength.)

4NancyLebovitz13y

I was basing my notion on having heard that a piano was a standard feature of a Victorian parlor. The original statement of the problem just specifies a piano, though I grant that the cartoon version requires a grand or baby grand. An upright piano just wouldn't be as funny. These days, there isn't any musical instrument which is a standard feature in the same way. Instead, being able to play recorded music is the standard. Thanks for the link about the lack of new musical instruments. I've been thinking for a while that stability of the classical orchestra meant there was something wrong, but it hadn't occurred to me that we've got the same stability in pop music.

3gwern13y

Sure, but think how small a fraction of the population that was. Most of Victorian England was, well, poor; coal miners or factory workers working 16 hour days, that sort of thing. Not wealthy bourgeoisie with parlors hosting the sort of high society ladies who were raised learning how to play piano, sketch, and faint in the arms of suitors. Unless it's set in a saloon! But given the low population density of the Old West, this is a relatively small error.

2David_Gerard13y

That article treats all forms of synthesis as one instrument. This is IMO not an accurate model. The explosion of electronic pop in the '80s was because the technology was on the upward slope of the logistic curve, and new stuff was becoming available on a regular basis for artists to gleefully seize upon. But even now, there's stuff you can do in 2013 that was largely out of reach, if not unknown, in 2000.

3satt13y

Have any handy examples? I find that a bit surprising (although it's a dead cert that you know more about pop music than I do, so you're probably right).

4David_Gerard13y

I'm talking mostly about new things you can do musically due to technology. The particular example I was thinking of was autotune, but that was actually invented in the late 1990s (whoops). But digital signal processing in general has benefited hugely in Moore's Law, and the ease afforded by being able to apply tens or hundreds of filters in real time. The phase change moment was when a musician could do this in faster than 1x time on a home PC. The past decade has been mostly on the top of the S-curve, though. Nevertheless, treating all synthesis as one thing is simply an incorrect model.

4satt13y

Funny coincidence. About a week ago I was telling someone that people sometimes give autotune as an example of a qualitatively new musical/aural device, even though Godley & Creme basically did it 30+ years ago. (Which doesn't contradict what you're saying; just because it was possible to mimic autotune in 1979 doesn't mean it was trivial, accessible, or doable in real time. Although autotune isn't new, being able to autotune on an industrial scale presumably is, 'cause of Moore's law.)

2zerker200013y

Granular synthesis is pretty fun.

0satt13y

Agreed, although I don't know how impractical or unknown it was in 2000 — I remember playing with GranuLab on my home PC around 2001.

5Larks13y

To 10m, surely?

4private_messaging13y

The workplace fatalities really gone down recently, with all the safe jobs of sitting in front of the computer. You should look for workplace fatalities in construction, preferably historical (before safety guidelines). Accounting for that would raise the estimate. A much bigger issue is that one has to actually stand under the piano as it is being lifted/lowered. The rate of such happening can be much (orders of magnitude) below that of fatal workplace accidents in general, and accounting for this would lower the estimate.

2gwern13y

I don't know where I would find them, and I'd guess that any reliable figures would be very recent: OSHA wasn't even founded until the 1970s, by which point there's already been huge shifts towards safer jobs. That was the point of going for lifetime risks, to avoid having to directly estimate per-lifting fatality rates - I thought about it for a while, but I couldn't see any remotely reasonable way to estimate how many pianos would fall and how often people would be near enough to be hit by it (which I could then estimate against number of pianos ever lifted to pull out a fatality rate, so instead I reversed the procedure and went with an overall fatality rate across all jobs).

0Decius13y

You also need to account for the fact that some proportion of piano-hoister work-related fatalities will be to other factors like heatstroke or heart attack or wrapping the rope around their arm.

4Kindly13y

To a very good first approximation, the distribution of falling piano deaths is Poisson. So if the expected number of deaths is in the range [0.39, 39], then the probability that no one has died of a falling piano is in the range [1e-17, 0.677] which would lead us to believe that with a probability of at least 1/3 such a death has occurred. (If 3.9 were the true average, then there's only a 2% chance of no such deaths.)

[-]gwern13y100

I disagree that the lower bound is 0; the right range is [-39,39]. Because after all, a falling piano can kill negative people: if a piano had fallen on Adolf Hitler in 1929, then it would have killed -5,999,999 people!

4Kindly13y

Sorry. The probability is in the range [1e-17, 1e17].

5Qiaochu_Yuan13y

That is a large probability.

1Decius13y

It's for when you need to be a thousand million billion percent sure of something.

2Mark_Eichenlaub13y

That approximation looks like this It'll overestimate by a lot if you do it over longer time periods. e.g. it overestimates this average by about 50% (your estimate actually gives 375, not 325), but if you went from 1m to 700m it would overestimate by a factor of about 3. A pretty-easy way to estimate total population under exponential growth is just current population 1/e lifetime. From your numbers, the population multiplies by e^2.5 in 300 years, so 120 years to multiply by e. That's two lifetimes, so the total number of lives is 700m2. For a smidgen more work you can get the "real" answer by doing 700m 2 - 50m 2.

[-]mwengler13y130

Cecil Adams tackled this one. Although he could find no documented cases of people being killed by a falling piano (or a falling safe), he did find one case of a guy being killed by a RISING piano while having sex with his girlfriend on it. What would you have estimated for the probability of that?

From the webpage:

The exception was the case of strip-club bouncer Jimmy Ferrozzo. In 1983 Jimmy and his dancer girlfriend were having sex on top of a piano that was rigged so it could be raised or lowered for performances. Apparently in the heat of passion the couple accidentally hit the up switch, whereupon the piano rose and crushed Jimmy to death against the ceiling. The girlfriend was pinned underneath him for hours but survived. I acknowledge this isn’t a scenario you want depicted in detail on the Saturday morning cartoons; my point is that death due to vertical piano movement has a basis in fact.

2A1987dM13y

Reality really is stranger than fiction, is it.

[-]TsviBT13y140

You did much better in Example #2 than you thought; the conclusion should read

60 fatalities per crash × 100 crashes with fatalities over the past 20 years = 6000 passenger fatalities from passenger-jet crashes in the past 20 years

which looks like a Fermi victory (albeit an arithmetic fail).

0lukeprog13y

Lol! Fixed.

[-]Oscar_Cunningham13y120

There are 3141 counties in the US. This is easy to remember because it's just the first four digits of pi (which you already have memorised, right?).

2[anonymous]13y

This reminds me of the surprisingly accurate approximation of pi x 10^7 seconds in a year.

1shaih13y

This chart has been extremely helpful to me in school and is full of weird approximation like the two above.

[-]Qiaochu_Yuan13y110

Thanks for writing this! This is definitely an important skill and it doesn't seem like there was such a post on LW already.

Some mild theoretical justification: one reason to expect this procedure to be reliable, especially if you break up an estimate into many pieces and multiply them, is that you expect the errors in your pieces to be more or less independent. That means they'll often more or less cancel out once you multiply them (e.g. one piece might be 4 times too large but another might be 5 times too small). More precisely, you can compute the variance of the logarithm of the final estimate and, as the number of pieces gets large, it will shrink compared to the expected value of the logarithm (and even more precisely, you can use something like Hoeffding's inequality).

Another mild justification is the notion of entangled truths. A lot of truths are entangled with the truth that there are about 300 million Americans and so on, so as long as you know a few relevant true facts about the world your estimates can't be too far off (unless the model you put those facts into is bad).

8pengvado13y

If success of a fermi estimate is defined to be "within a factor of 10 of the correct answer", then that's a constant bound on the allowed error of the logarithm. No "compared to the expected value of the logarithm" involved. Besides, I wouldn't expect the value of the logarithm to grow with number of pieces either: the log of an individual piece can be negative, and the true answer doesn't get bigger just because you split the problem into more pieces. So, assuming independent errors and using either Hoeffding's inequality or the central limit theorem to estimate the error of the result, says that you're better off using as few inputs as possible. The reason fermi estimates even involve more than 1 step, is that you can make the per-step error smaller by choosing pieces that you're somewhat confident of.

2Qiaochu_Yuan13y

Oops, you're absolutely right. Thanks for the correction!

[-]Morendil13y90

Tip: frame your estimates in terms of intervals with confidence levels, i.e. "90% probability that the answer is within and ". Try to work out both a 90% and a 50% interval.

I've found interval estimates to be much more useful than point estimates, and they combine very well with Fermi techniques if you keep track of how much rounding you've introduced overall.

In addition, you can compute a Brier score when/if you find out the correct answer, which gives you a target for improvement.

0Heka13y

Douglas W. Hubbard has a book titled How to Measure Anything where he states that half a day of exercising confidence interval calibration makes most people nearly perfectly calibrated. As you noted and as is said here, that method fits nicely with Fermi estimates. This combination seems to have a great ratio between training time and usefulness.

[-]TitaniumDragon13y70

I will note that I went through the mental exercise of cars in a much simpler (and I would say better) way: I took the number of cars in the US (300 million was my guess for this, which is actually fairly close to the actual figure of 254 million claimed by the same article that you referenced) and guessed about how long cars typically ended up lasting before they went away (my estimate range was 10-30 years on average). To have 300 million cars, that would suggest that we would have to purchase new cars at a sufficiently high rate to maintain that number ... (read more)

[-][anonymous]13y60

Alternatively, you might allow yourself to look up particular pieces of the problem — e.g. the number of Sikhs in the world, the formula for escape velocity, or the gross world product — but not the final quantity you're trying to estimate.

Would it bankrupt the global economy to orbit all the world's Sikhs?

[This comment is no longer endorsed by its author]Reply

[-]lukeprog13y50

1Thej Kiran4y

http://t.www.na-businesspress.com/JABE/Jabe105/AndersonWeb.pdf I think this might be the updated link.

[-]lukeprog13y50

Write down your own Fermi estimation attempts here. One Fermi estimate per comment, please!

6lukeprog13y

One famous Fermi estimate is the Drake equation.

6gwern13y

A running list of my own: http://www.gwern.net/Notes#fermi-calculations (And there's a number of them floating around predictionbook.com; Fermi-style loose reasoning is great for constraining predictions.)

5Qiaochu_Yuan13y

Here's one I did with Marcello awhile ago: about how many high schools are there in the US? My attempt: there are 50 states. Each state has maybe 20 school districts. Each district has maybe 10 high schools. So 50 20 10 = 10,000 high schools. Marcello's attempt (IIRC): there are 300 million Americans. Of these, maybe 50 million are in high school. There are maybe 1,000 students in a high school. So 50,000,000 / 1,000 = 50,000 high schools. Actual answer: ... ... ... Numbers vary, I think depending on what is being counted as a high school, but it looks like the actual number is between 18,000 and 24,000. As it turns out, the first approach underestimated the total number of school districts in the US (it's more like 14,000) but overestimated the number of high schools per district. The second approach overestimated the number of high school students (it's more like 14 million) but also overestimated the average number of students per high school. And the geometric mean of the two approaches is 22,000, which is quite close!

7Kindly13y

I tried the second approach with better success: it helps to break up the "how many Americans are in high school" calculation. If the average American lives for 80 years, and goes to high school for 4, then 1/20 of all Americans are in high school, which is 15 million.

3ArisKatsaris13y

...you guessed 1 out of 6 Americans is in highschool? With an average lifespan of 70+ years and a highschool duration of 3 years (edit: oh, it's 4 years in the US?), shouldn't it be somewhere between 1 in 20 and 1 in 25?

4Qiaochu_Yuan13y

This conversation happened something like a month ago, and it was Marcello using this approach, not me, so my memory of what Marcello did is fuzzy, but IIRC he used a big number. The distribution of population shouldn't be exactly uniform with respect to age, although it's probably more uniform now than it used to be.

4Ronak13y

Just tried one today: how safe are planes? Last time I was at an airport, the screen had five flights, three-hour period. It was peak time, so multiplied only by fifteen, so 25 flights from Chennai airport per day. ~200 countries in the world, so guessed 500 adjusted airports (effective no. of airports of size of Chennai airport), giving 12500 flights a day and 3*10^6 flights a year. One crash a year from my news memories, gives possibility of plane crash as 1/310^-6 ~ 310^-7. Probability of dying in a plane crash is 3*10^-7 (source). At hundred dead passengers a flight, fatal crashes are ~ 10^-5. Off by two orders of magnitude.

4CCC13y

If there are 310^6 flights in a year, and one randomly selected plane crashes per year on average, with all aboard being killed, then the chances of dying in an airplane crash are %20\10%5E{-6}), surely? Yes, there's a hundred dead passengers on the flight that went down, but there's also a hundred living passengers on every flight that didn't go down. The hundreds cancel out.

0Ronak13y

Wow, that was stupid of me. Of course they do! And thanks.

3lukeprog13y

Anna Salamon's Singularity Summit talk from a few years ago explains one Fermi estimate regarding the value of gathering more information about AI impacts: How Much it Matters to Know What Matters: A Back of the Envelope Calculation.

2mwengler13y

How old is the SECOND oldest person in the world compared to the oldest? Same for the united states? I bogged down long before I got the answer. Below is the gibberish I generated towards bogging down. So OK, I don't even know offhand how old is the oldest, but I would bet it is in the 114 years old (yo) to 120 yo range. Then figure in some hand-wavey way that people die at ages normally distributed with a mean of 75 yo. We can estimate how many sigma (standard deviations) away from that is the oldest person. Figure there are 6 billion people now, but I know this number has grown a lot in my lifetime, it was less than 4 billion when I was born 55.95 years ago. So say the 75 yo's come from a population consistent with 3 billion people. 1/2 die younger than 75, 1/2 die older, so the oldest person in the world is 1 in 1.5 billion on the distribution. OK what do I know about normal distributions? Normal distribution goes as exp ( -((mean-x)/(2sigma))^2 ). So at what x is exp( -(x/2sigma)^2 ) = 1e-9? (x / 2sigma) ^ 2 = -ln ( 1e-9). How to estimate natural log of a billionth? e = 2.7 is close enough for government work to the sqrt(10). So ln(z) = 2log_10(z). Then -ln(1e-9) = -2log_10(1e-9) = 29 = 18. So (x/2sigma)^2 = 18, sqrt(18) = 4 so So I got 1 in a billion is 4 sigma. I didn't trust that so I looked that up, Maybe I should have trusted it, in fact 1 in a billion is (slightly more than ) 6 sigma. mean of 75 yo, x=115 yo, x-mean = 40 years. 6 sigma is 40 years. 1 sigma=6 years. So do I have ANYTHING yet? I am looking for dx where exp(-((x+dx)/(2sigma))^2) - exp( -(x/2sigma)^2)

[-]Vaniver13y140

So, this isn't quite appropriate for Fermi calculations, because the math involved is a bit intense to do in your head. But here's how you'd actually do it:

Age-related mortality follows a Gompertz curve, which has much, much shorter tails than a normal distribution.

I'd start with order statistics. If you have a population of 5 billion people, then the expected percentile of the top person is 1-(1/10e9), and the expected percentile of the second best person is 1-(3/10e9). (Why is it a 3, instead of a 2? Because each of these expectations is in the middle of a range that's 1/5e9, or 2/10e9, wide.)

So, the expected age* of death for the oldest person is 114.46, using the numbers from that post (and committing the sin of reporting several more significant figures), and the expected age of death for the second oldest person is 113.97. That suggests a gap of about six months between the oldest and second oldest.

* I should be clear that this is the age corresponding to the expected percentile, not the expected age, which is a more involved calculation. They should be pretty close, especially given our huge population size.

But terminal age and current age are different- it could actually be... (read more)

5A1987dM13y

Upvoted (among other things) for a way of determining the distribution of order statistics from an arbitrary distribution knowing those of a uniform distribution which sounds obvious in retrospect but to be honest I would never have come up with on my own.

1Shmi13y

Assuming dx << x, this is approximated by a differential, (-xdx/sigma^2) * exp( -(x/2sigma)^2, or the relative drop of dx/sigma^2. You want it to be 1/2 (lost one person out of two), your x = 4 sigma, so dx=1/8 sigma, which is under a year. Of course, it's rather optimistic to apply the normal distribution to this problem, to begin with.

2Heka13y

I estimated how much the population of Helsinki (capital of Finland) grew in 2012. I knew from the news that the growth rate is considered to be steep. I knew there are currently about 500 000 habitants in Helsinki. I set the upper bound to 3 % growth rate or 15 000 residents for now. With that rate the city would grow twentyfold in 100 years which is too much. But the rate might be steeper now. For lower bound i chose 1000 new residents. I felt that anything less couldnt really produce any news. AGM is 3750. My second method was to go through the number of new apartments. Here I just checked that in recent years about 3000 apartments have been built yearly. Guessing that the household size could be 2 persons I got 6000 new residents. It turned out that the population grew by 8300 residents which is highest in 17 years. Otherwise it has recently been around 6000. So both methods worked well. Both have the benefit that one doesnt need to care whether the growth comes from births/deaths or people flow. They also didn't require considering how many people move out and how many come in. Obviously i was much more confident on the second method. Which makes me think that applying confidence intervals to fermi estimates would be useful.

0ParanoidAltoid13y

For the "Only Shallow" one, I couldn't think of a good way to break it down, and so began by approximating the total number of listens at 2 million. My final estimate was off by a factor of one.

0lukeprog13y

Matt Mahoney's estimate of the cost of AI is a sort-of Fermi estimate.

0Furslid13y

Out of the price of a new car, how much goes to buying raw materials? How much to capital owners? How much to labor?

0Dr_Manhattan13y

How many Wall-Marts in the USA.

3Nornagest13y

That sounds like the kind of thing you could just Google. But I'll bite. Wal-Marts have the advantage of being pretty evenly distributed geographically; there's rarely more than one within easy driving distance. I recall there being about 15,000 towns in the US, but they aren't uniformly distributed; they tend to cluster, and even among those that aren't clustered a good number are going to be too small to support a Wal-Mart. So let's assume there's one Wal-Mart per five towns on average, taking into account clustering effects and towns too small or isolated to support one. That gives us a figure of 3,000 Wal-Marts. When I Google it, that turns out to be irel pybfr gb gur ahzore bs Jny-Zneg Fhcrepragref, gur ynetr syntfuvc fgberf gung gur cuenfr "Jny-Zneg" oevatf gb zvaq. Ubjrire, Jny-Zneg nyfb bcrengrf n fznyyre ahzore bs "qvfpbhag fgber", "arvtuobeubbq znexrg", naq "rkcerff" ybpngvbaf gung funer gur fnzr oenaqvat. Vs jr vapyhqr "qvfpbhag" naq "arvtuobeubbq" ybpngvbaf, gur gbgny vf nobhg guerr gubhfnaq rvtug uhaqerq. V pna'g svaq gur ahzore bs "rkcerff" fgberf, ohg gur sbezng jnf perngrq va 2011 fb gurer cebonoyl nera'g gbb znal.

2Furslid13y

Different method. Assume all 300 million us citizens are served by a Wal Mart. Any population that doesn't live near a Wal-Mart has to be small enough to ignore. Each Wal-mart probably has between 10,000 and 1 million potential customers. Both fringes seem unlikely, so we can be within a factor of 10 by guessing 100000 people per Wal-Mart. This also leads to 3000 Wal-Marts in the US.

[-]Friendly-HI13y40

Fermi estimates can help you become more efficient in your day-to-day life, and give you increased confidence in the decisions you face. If you want to become proficient in making Fermi estimates, I recommend practicing them 30 minutes per day for three months. In that time, you should be able to make about (2 Fermis per day)×(90 days) = 180 Fermi estimates.

I'm not sure about this claim about day-to-day life. Maybe there are some lines of work where this skill could be useful, but in general it's quite rare in day-to-day life where you have to come up w... (read more)

[-]lmm13y30

The guys at last.fm are usually very willing to help out with interesting research (or at least were when I worked there a couple of years ago), so if you particularly care about that information it's worth trying to contact them.

[-]Thej Kiran4y20

Play Fermi Questions: 2100 Fermi problems and counting.

http://www.fermiquestions.com/ link doesn't work anymore. It is also not on wayback machine. :(

[-]Shmi13y20

You can try estimating, or keep listing the reasons why it's hard.

[-]mwengler13y20

One of my favorite numbers to remember to aid in estimations is this: 1 year = pi * 10^7 seconds. Its really pretty accurate.

Of course for Fermi estimation just remember 1 Gs (gigasecond) = 30 years.

0A1987dM13y

10^7.5 is even more accurate.

[-]jojobee13y20

I spend probably a pretty unusual amount of time estimating things for fun, and have come to use more or less this exact process on my own over time from doing it.

One thing I've observed, but haven't truly tested, is my geometric means seem to be much more effective when I'm willing to put a more tight guess on them. I started off bounding them with what I thought the answer conceivably could be, which seemed objective and often felt easier to estimate. The problem was that often either the lower or upper bound was too arbitrary relative to it's weight on ... (read more)

[-]keltan2y10

To help remember this post and it's methods I broke it down into song lyrics and used Udio to make the song.

[-]fermilover5y1-1

Check out https://www.fermiproblems.in/ for a great collection of guesstimates / fermiproblems

[-]son0fhobs13y10

I just wanted to say, after reading the Fermi estimate of cars in the US, I literally clapped - out loud. Well done. And I highly appreciate the honest poor first attempt - so that I don't feel like such an idiot next time I completely fail.

[-]John_Maxwell13y10

Potentially useful: http://instacalc.com/

[-]gothgirl42066613y10

I'm happy to see that the Greatest Band of All Time is the only rock band I can recall ever mentioned in a top-level LessWrong post. I thought rationalists just sort of listened only to Great Works like Bach or Mozart, but I guess I was wrong. Clearly lukeprog used his skills as a rationalist to rationally deduce the band with the greatest talent, creativity, and artistic impact of the last thirty years and then decided to put a reference to them in this post :)

7NancyLebovitz13y

If you check out Media posts, you'll see that LWers like a range of music. It wouldn't surprise me too much if they tend to like contemporary classical better than classical classical.

0ikrase13y

I like a specific subset of classical classical, but I suspect not at all typical.

3Qiaochu_Yuan13y

Why?

3Furslid13y

Because of the images of different musical genres in our culture. There is an association of classical music and being academic or upper class. In popular media, liking classical music is a cheap signal for these character types. This naturally triggers confirmation biases, as we view the rationalist listening to Bach as typical, and the rationalist listening to The Rolling Stones as atypical. People also use musical preference to signal what type of person they are. If someone wants to be seen as a rationalist, they often mention their love of Bach and don't mention genres with a different image, except to disparage them.

8Qiaochu_Yuan13y

I think you're conflating "rationalist" and "intellectual." I agree that there is a stereotype that intellectuals only listen to Great Works like Bach or Mozart, but I'm curious where the OP picked up that this stereotype also ought to apply to LW-style rationalists. I mean, Eliezer takes pains in the Sequences to make anime references specifically to avoid this kind of thing.

5[anonymous]13y

Well, he also likes tends to like anime, and anime has a tendency to deal with some future-ish issues.

[-]Qiaochu_Yuan13y140

From On Things That Are Awesome:

Whenever someone compliments "Eliezer Yudkowsky", they are really complimenting "Eliezer Yudkowsky's writing" or "Eliezer Yudkowsky's best writing that stands out most in my mind". People who met me in person were often shocked at how much my in-person impression departed from the picture they had in their minds. I think this mostly had to do with imagining me as being the sort of actor who would be chosen to play me in the movie version of my life—they imagined way too much dignity. That forms a large part of the reason why I occasionally toss in the deliberate anime reference, which does seem to have fixed the divergence a bit.

1Furslid13y

I'm just pointing out the way such a bias comes into being. I know I don't listen to classical, and although I'd expect a slightly higher proportion here than in the general population, I wouldn't guess it wold be a majority or significant plurality. If I had to guess, I'd guess on varied musical tastes, probably trending towards more niche genres than broad spectrum pop than the general population.

0[anonymous]13y

1gothgirl42066613y

Well, Eliezer mentions Bach a bunch in the sequences as an example of a great work of art. I used stereotypes to extrapolate. :p (the statement was tongue-in-cheek, if that didn't come across. I am genuinely a little surprised to see MBV mentioned here though.)

[-]CCC13y00

I recently ran across an article describing how to find a rough estimate of the standard deviation of a population, given a number of samples, which seems that it would be suitable for Fermi estimates of probability distributions.

First of all, you need a large enough population that the central limit theorem applies, and the distribution can therefore be assumed to be normal. In a normal distribution, 99.73% of the samples will be within three standard deviations of the mean (either above or below; a total range of six standard deviations). Therefore, one ... (read more)

3Kindly13y

If you want a range containing 70% of the samples, why wouldn't you just take the range between the 15th and 85th percentile values?

1CCC13y

That would also work, and probably be more accurate, but I suspect that it would take longer to find the 15th and 85th percentile values than it would to find the ends of the range.

[-]Shmi13y00

How long can the International Space Station stay up without a boost? I can think of a couple of ways to estimate that.

[-]Furslid13y00

Out of the price of a new car, how much goes to buying raw materials? How much to capital owners? How much to labor?

[This comment is no longer endorsed by its author]Reply

[-]MichaelVassar13y-10

I recommend trying to take the harmonic mean of a physical and an economic estimate when appropriate.

[-]gjm13y210

I recommend doing everything when appropriate.

Is there a particular reason why the harmonic mean would be a particularly suitable tool for combining physical and economic estimates? I've spent only a few seconds trying to think of one, failed, and had trouble motivating myself to look harder because on the face of it it seems like for most problems for which you might want to do this you're about equally likely to be finding any given quantity as its reciprocal, which suggests that a general preference for the harmonic mean is unlikely to be a good strategy -- what am I missing?

9SarahNibs13y

Can I have a representative example of a problem where this is appropriate?

2BlazeOrangeDeer13y

So, what you're saying is that the larger number is less likely to be accurate the further it is from the smaller number? Why is that?

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Estimation tips

Fermi estimation failure modes

Fermi practice

Example 1: How many new passenger cars are sold each year in the USA?

Approach #1: Car dealerships

Approach #2: Population of the USA

Example 2: How many fatalities from passenger-jet crashes have there been in the past 20 years?

Example 3: How much does the New York state government spends on K-12 education every year?

Example 4: How many plays of My Bloody Valentine's "Only Shallow" have been reported to last.fm?

Further examples

Conclusion