[ Question ]

Fermi Challenge: Trains and Air Cargo

by Ben Pace1 min read5th Oct 202021 comments


Exercises / Problem-SetsRationalityPracticalWorld Modeling

Jacob (Jacob) and I are giving each other Fermi estimates to complete.

  1. From Ben: How many miles of train tracks exist in the world?
  2. From Jacob: How many metric tons of air freight cargo shipped globally in 2009, and in 2019 (either by commercial airlines or specialized freight companies)? (Final score will be average of those scores.

Whoever is the furthest from the truth (in log space) loses. 

The deadline is Monday 12th October. You are welcome to participate.

If you submit a Fermi estimate as an answer (not comment) to this post, it will be scored. If you submit for both and would like them both to be scored, your scores will be averaged.

This thread has the following rules which you must obey or else your answers/comments will be edited/deleted.

Rules for this thread

  1. All analysis of the question must be in spoiler tags (how do I do that?)
  2. You're only allowed to use numbers you don't need to Google.
  3. No object level analysis is allowed outside spoilered text.
  4. If your comment contains the answer, or if you know the true answer, make sure to mention this outside the spoilered text.

If you'd like an example of what that looks like in practise, see this and this thread. The first thread is especially good, where only whitelisted things are allowed outside of the spoiler text.

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Why do this?

Last week Jacob ran the babble challenge. More than 20 users joined to practice their creativity by coming up with 50 ways of sending something to the moon. 

This challenge continues in the same spirit. We want to build a culture of practice on LessWrong. A place where you users can come together, push the limits of their abilities, and grow stronger as rationalists. This is an experiment in that direction. 


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9 Answers

True answer to my question:

Got the data from Statista: https://www.statista.com/statistics/564668/worldwide-air-cargo-traffic/

2009: 42.3M metric tons. 
2019: 61.3M metric tons.

Rankings of participants (spreadsheet here): 

Participant (absolute distance from truth in log space)
Vanilla_cabs (0.1)
Magfrump (0.14)
knite (0.21)
philh (0.23)
smiley314 (0.48)

Attempt at q1:

The earth has a radius of 6400 km. Surface area is 4πr² and about 1/3 of that is land, giving around 200 million square kilometers of land surface.

I think... most of that is barely populated, and probably has few train tracks? Let's say 1% of it is densely popualated, 10% is sparsely populated, and 90% is basically unpopulated. In the densely populated bit, I could believe 1 km track per square km area. In the sparsely populated let's say a tenth of that, and ignore the rest. That gives... 2 million km in dense, plus another 2 million in sparse for 4 million km in total.

That feels low, I think? But I'll stick with it unless I come up with something better.


I think a lot more freight goes by boat then plane. Let's say plane is 1% of boat.

I think an aircraft carrier displaces, what, 100,000 metric tons? So it's maybe reasonable to guess that a respectable bulk transport can carry 100,000 metric tons of cargo.

Let's say at any given time there are 100 of those underway, on journeys lasting 30 days. That makes about 100,000,000 metric tons shipped annually by boat, and 1,000,000 by plane.

Between 2009 and 2019 I'm gonna guess it went up by enough to count as one order of magnitude. So let's split the difference and call it 300,000 in 2009 and 3,000,000 in 2019.

2philh3moAnother attempt at q2: Suppose air freight has dectupled every decade, starting at one metric ton in 1909. Then we get 10^10 metric tons in 2009 and 10^11 in 2019. That's 4½ orders of magnitude more than my other answer. :/ I currently suspect this one is too high and that one is too low, but that one is closer.

Hi everyone! Very, very, very low confidence – these are my two very first Fermi estimates ever; so Feedback will be very much appreciated.


Let‘s assume that every train station has ten rail tracks and that every rail track at a train station is ten miles long. That makes 100 miles for every train station. How many train stations are there in the world – 10^6 (1.000.000)? Seems a bit low, let‘s go with 10^9. So that‘s 10^2 * 10^9 = 10^11 miles only for train stations.

So how long are train rails that actually connect train stations combined? Uh… Let‘s assume that there will always be two train tracks for every connection (as for both directions, one track per each direction). Let‘s assume average length 100 miles for one direction. How many connections are there? Let‘s say 10^9 (I thought, it might make sense to have around as much connections as stations), so that makes 10^9 * 10^2 * 2 = 210^11 miles. Together with our previous result that gets us
10^11 + 10^11 = 3*10^11 = 300.000.000.000 miles. That seems a bit large, but hey… First try ever.


I have absolutely no idea, and so randomly guess 10.000 metric tons in 2009 and a 10% increase if we compare 2019 with 2009. That makes 10.000+1.000=11.000 metric tons for 2019. Taking the average gives us (11.000 – 10.000)/2 = 1.000/2 = 500 metric tons as final result; at least if I interpret the task assignment correctly.

Best regards, smiley314

Congrats on making your first Fermi! You are taking the first step on and a long and rewarding journey of quantitative rationality :) 

As for q1, claiming that the world has a billion train stations seems way too high to me. That would be enough for every two families to have "their own" train station. But when you look at the throughput of passengers there's 1000+ at least per station. 

I also think you're neglecting the fact that train stations are connected to each other, and so end up double counting some tracks?


UK railways are I estimate ~6,000 miles

  • Main routes ~2,000 miles 
    • UK is ~600 miles long and 150 miles wide
    • Multiple north-south and east-west lines
  • Minor routes ~4,000 miles
    • Not sure how to estimate these and ended up just doubling the main routes

Assuming miles of railway per capita in the UK is typical of the world this implies 700k miles of lines.

(Miles of railways per land area seems less accurate - e.g. Siberia will have loads of area but relatively few miles of railway.)

I guess the UK is actually higher than average. I think China, Japan, Europe and India have lots of railways but even then possibly less railways per capita. I think US, South America and Africa less.

I'm just going to halve it as most of the effect per capita will be driven by China and India both of which I guess are in that region.

So: 350k miles

Argh I'm an idiot I just read magfrump's estimate and realised I forgot to include that most lines are doubles so double my estimate back to 700k! I know, that's cheating so accept my first answer.

A country with trains might have the equivalent of 3-10 times the length of the country worth of train tracks, and countries are roughly 600-3000 miles across, with perhaps 10-100 countries with a lot of trains that are also not far below the 600 mile size. The geometric means are approximately 5.5vlength , 1350 miles, 30 countries. Multiply all of these and you get 222,750 miles.

Q1 Answer: 222,750 miles.

Post-hoc edit: If you take these numbers and do the right math on them, you get a more accurate answer. The right math for this kind of thing is not just to multiply the geometric means. You want to take 50 draws from a random variable that's the product of size S times length multiplier L, which are lognormally distributed. If you do that you get about 600,000. If you do it for 30, you get 346,000. So, if you use the numbers I gave with some actual math, it's 346,000. But I looked at the correct answer before deciding to bother to do the math.

Air freight is perhaps 10%-30% of passenger travel since I see more passenger planes at airports than freight. Passenger travel is easier to estimate. Say that each of 1,000,000,000 people travel twice per year by air. 15 people plus their baggage is one metric ton. 2 × 17% × 1,000,000,000 / 15 = 22,000,000 metric tons.

Q2 Answer: 22,000,000 metric tons per year for both years.

Well, better a rough model than no model!

I took the geometric mean of three models implemented in Guesstimate, here

One tried to work from how many people travel by train each day, one from the circumference of the earth, and one from how many packages are shipped each year. 

Didn't use Google for anything, assuming that was against the rules. 

This is probably one of the weirdest guesstimates I've made in terms of approach. I'm sure there's something easier. 

Anyway, my final answer is 420 000 km.

Trying this spending five (turned into 12) minutes or less per question


 US is o(10k) miles across, probably similar for Europe, China. I have no idea whether Africa or South America have significant train tracks. I know there are only a few transcontinental lines on the US, probably similar for Europe and China, so maybe 100k miles of long range east-west track. north-south is probably a bit less, call it 160k miles of long distance tracks. I live in train-impoverished California, where there are probably fewer train tracks than in other places, no idea whether that's biased toward long distance or local though. Near here, there's a train that at least goes to Sacramento, call it about 100 miles of longer-distance train. Then within the bay there's caltrain and light rail and bart, which creates maybe two circles around the bay, which is like 30 miles n/s and maybe 10 e/w, so a little less than 100 miles around, so my current guess is local trains have about the same amount of track as long distance trains. long distance tracks are probably bi-directional, short range are more likely to be more than that, but not everywhere. Maybe call it 3 tracks per mile. Then 160k miles x (2 tracks of long distance + 3 local tracks) is 800k miles. I don't like this estimate because it's very conjunctive but I'll leave it up anyway.


The number of planes flying between international hubs commercially is probably not that far off from the number of cargo planes flying internationally, otherwise you could inexpensively increase the number of passenger flights or amount of freight by adding some more of the other to existing flights. Probably that's not a perfect tradeoff so probably I'll adjust the freight numbers up after. Major flight hubs probably are in china/japan, west coast us, east coast us, europe. Maybe 2-3 flights per day between any pair of intercontinental hubs. So asia <-> west coast there are maybe 3-5 hubs x 3-5 hubs = 9-25 flights per day, east coast to europe again, and probably I'm missing something so lets double that for ~60 hub pairs x 2-3 flights x 2x that much freight is 300 airplane-fulls of freight. A passenger airplane IME has ~30 rows of ~6-10 seats, probably with the top half of the plane populated, for a volume of 120ft length time 30ft diameter circle is 720 sqft, so 75k cubic feet. Cargo is probably a bit less dense than water, 75k cubic feet is ~3k cubic meters, a milliliter is a cubic millimeter so that's 3e12 ml3 = 3e9 L and a liter of water weighs 1kg, that's WAY too much weight so let's try estimating person weight instead. 300 people *100kg + 50kg baggage = 45000 kg, 45 tons times 300 airplane-fulls is 13k tons/day so 4 million tons/year. I haven't distinguished at all between 2019 and 2009. Obviously 2019 will be more, probably mostly in and out of China. Since China accounts for ~1/6 of my calculation let's call it 1/6 less in 2009 and 1/6 more in 2019, which is completely unjustified reasoning but I've gone way over my time limit so 2009: 3.3 million tons, 2019 4.7 million tons.

How many miles of train tracks exist in the world?

Radius of Earth = ~3,000 miles

Surface area of earth = 4 x pi x r^2 = 1.13 × 10^8 = ~10^8

25% of the earth is land 

10% of the land "needs" tracks

.1% of that land actually is tracks

10^8 x .25 x .1 x .001 = 2,500 miles

This feels obviously wrong, so let's buff the numbers:

Radius = 5,000 miles, 30% of the surface is land, 25% "needs" track, .5% of the land is tracks

3.14 x 10^8 x .35 x .2 x .005 = ~110,000 miles

Let's round that off to 100,000 miles.


Edit: Thanks to the comment from @philh, I see that I calculated square miles rather than linear miles. Assuming the width of a track is 1/1000 of a mile:

2500 x 1000 = ~2.5 million miles

This feels reasonable, so if I had performed that step, I would not have done a second round of estimation.



How many metric tons of air freight cargo shipped globally in 2009, and in 2019 (either by commercial airlines or specialized freight companies)?

How much stuff did I order? Everything I use was shipped from somewhere.

Amount of stuff I ordered last year: 200 pounds

That feels low when I think about heavy food and drink containers, so let's bump to 1,000 pounds

My share of infrastructure "stuff" feels smaller than that, so no modifier.

Most of the world uses less stuff than I do (age, affluence), so nudge it down to 700 pounds.

A ton is 2,000 pounds

Population of the world: 7 billion

7 billion * 700 / 2,000 = ~2.5 billion

Let's assume 2% growth per year

1.02^10 = 1.22

2.5 billion / 1.22 = ~2 billion


2019: 2.5 billion tons

2009: 2 billion tons


After reading the answers from others, I've realized that I assumed all freight was by air. If I had considered that, I would apply a discount factor, revising significantly downward. Keeping my original answer in the spirit of the exercise.

Comment on q1:

It looks like your calculations are giving you square miles of track. If a track is 1/1000 of a mile wide (1.6 meters? sure, close enough, judging by the height of a damsel in distress), you'd have 2.5 million linear miles from your first estimate, and 100 million linear miles from your second.

1knite3moThat's a great catch, thanks! With that correction as a final step, I feel much better about my initial estimate!

Low confidence, but let's play anyways:


5 million miles


For both 2009 and 2019: 300 million tons

By the way, for q2, what is considered cargo? Are passengers considered cargo? How about the crew? How about hand luggage of the passengers? Hand luggage of the crew? The meals and other consumables? (my answer assumed no for all these questions)

6 comments, sorted by Highlighting new comments since Today at 10:06 PM

Meta note: Spoilers don’t function correctly on compressed comments on the front page so you get to see the first few words. Generally this isn’t a problem but I can imagine there would be times that it would.

The butler did it

Related to this question, I discovered a new rule for Fermi estimates.

If you want to estimate the mean value of a lognormally distributed random variable, giving the middle order of magnitude will be wrong as these are skewed.

There is a simple rule for getting this right that I discovered: take your middle order of magnitude (i.e. if you think it's between 10 and 100, the middle is 10^1.5) and add 1.15 times the square of your estimate of the standard deviation in log-space. So in this case that's something 1.5 +1.15 × 0.5^2. Then do 10 to the power that. This gives you about 60 - twice the answer you would have gotten with the middle order of magnitude.

This applies to things like "how big is a country" and "how many miles of track is there for a country of a given size" which might both be lognormally distributed.


How much research is appropriate for generating a the estimates? In principle I feel like I should do my estimate off-the-cuff, but there are many pieces of info that would probably be helpful for intermediate estimates. But in the absence of enforcement mechanisms or meaningful distinctions between looking up subproblems of various size vs. the whole answer, I'd be worried that not looking anything up at all would lead to losing.

I forgot to include the norms. The rule is that you're only allowed to use numbers you don't need to Google. I've added it to the thread rules.

If I have specific knowledge (counts and measurements) about the passenger and freight rail industries, does that count as "object level"?

Yes! Please spoiler your answer.