8.2 Fermi Problems

Physicists train their students in doing "Fermi problems," back-of-the-envelope estimates of quantities that arise in physical problems and in life. This is useful as an approach to performing "sanity checks" of claims in the world and of your own ideas and beliefs. Checking numbers with quick Fermi estimates may be even more important in a world in which it is difficult to evaluate the credibility of numbers quoted in news articles or social media posts.

The Lesson in Context

It is often important to have a rough idea about the size of a number for the purpose of decision making. Even if the quantity is difficult to immediately visualize, it is often possible to estimate it by multiplying smaller numbers that we do have an idea about, a technique called Fermi estimation. In this lesson, we walk students through a couple of simple Fermi problems and give them the opportunity to solve new ones on their own.

Earlier Lessons

8.1 Orders of Understanding

• For causal problems that can be quantified, e.g. carbon emissions, water usage, budget, Fermi estimation is often a good way to compare the order of importance of different causes.

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Later Lessons

14.1 Scenario Planning

• When planning for future scenarios, one can make rough Fermi estimates for the magnitude of the impact of each scenario.

Takeaways

Learning GoalsDefinitionsExamplesExpanded Learning Goals

After this lesson, students should

1. Be confident in their ability to make a reasonable magnitude estimate of quantities for which they have no direct knowledge.
2. Identify quantities that would and would not be appropriate to estimate with a Fermi calculation.
3. Provide rough estimates for real-world quantities using "back-of-the-envelope" (Fermi) approximations.
4. Evaluate the credibility of quantitative statements using "back-of-the-envelope" approximations.
5. Use Fermi estimates to identify first, second, third order causes for example problems, and estimate their effect sizes.

Fermi Estimate

A systematic estimate of a quantity based on what you know. The typical goal is to get within an order of magnitude of the right answer. (This often proves possible even for topics about which you know very little.) The steps to do this are the following.

1. Decompose the problem into multiple components that you can estimate. (Break down unfamiliar components into familiar components).
2. Estimate components using approximations.
3. Combine estimated components to calculate Fermi estimate.
4. Optional: Compute upper and lower bounds (maximum and minimum quantities above/below between which you are fairly confident the correct estimate should be).

Exemplary Quotes

“He's suggesting that the Federal budget deficit is due to the money we spend on job training programs. But that's ridiculous! Even if every single person out of work—let's imagine that it is 10% of the working-age population (say 10 million people out of work)—went to a job training program that cost as much as a year of college at a good university (say, $40,000), that would cost 400 billion dollars. Hmm... well that's not quite as small as I expected, but it's still not trillions of dollars, and furthermore I am sure we aren't spending that much on each person for job training. Let see, can I estimate that cost per person in some more realistic way than using college costs...”

“There are about 1500 students in the introductory to computer science class at UC Berkeley. Each final is about 10 printed pages long. A rough guess of the cost of printing one page for bulk orders like this is probably around $0.05 per page. That means the University spends around $750 in printing fees to administer one test.”

Source

“Claim: The university profits on each college application they receive, so are incentivized to get as many people to apply even though they know they won't accept them. Estimate: The application fee for UC Berkeley is $70. It probably takes an hour total for every admissions officer to read an application and the committee to agree. Let's say the average salary of an admissions officer is $100,000 a year. If they work for 50 weeks a year for 40 hours a week that means their average salary is about $50 an hour. That would mean that the university makes a seemingly sizable profit on each application.”

“In American journalist Richard Harding Davis' account of the German Army's march through Brussels in the first World War he claims that the German army marched unbroken for "three days and three nights" through the city. Is this believable? 3 days and 3 nights is 72 hours. Let's say the average marching speed of a soldier is 5 mph. That would mean the German army stretched backwards for 5 mph * 72 h ＝ 360 miles backwards from the city. Let's say the Germans march in columns two soldiers wide. Each row probably has about 3 feet of space in front of it. 360 miles * 5280 feet per mile ≃ 1.9 million feet. 1.9 million feet * 2 soldiers / 3 feet ≃ 1.27 million soldiers. The actual size of the German army in the first World War numbered in the millions and the main force passed through Belgium to reach France, so this number is very plausible.”

Source

“Question: How much do you think a 747 weighs? "A car fits about 5 people and weighs about 2 tons. A 747 fits maybe 500 people. Although airliners have much longer range than cars they are also designed to be much lighter because they have to fly. Carrying 100x more people therefore might make the weight of a 747 about 200 tons. The actual empty weight of a 747 is about 203 tons."”

Source

After this lesson, students should

1. Attitudes
   1. Be confident in one's ability to make a reasonable magnitude estimate of quantities for which one has no intuitive guess or direct knowledge.
2. Concept Acquisition
   1. Fermi Estimates: A systematic estimate of a quantity based on what you know. The typical goal is to get within an order of magnitude of the right answer. (This often proves possible even for topics about which you know very little.)
      1. Decompose the problem into multiple components that you can estimate. (Break down unfamiliar components into familiar components.)
      2. Estimate components using approximations.
      3. Combine estimated components to calculate the Fermi estimate.
      4. Compute upper and lower bounds (maximum and minimum quantities above/below between which you are fairly confident the correct estimate should be).
   2. Order of Magnitude: Factor of ten.
3. Concept Application
   1. Identify quantities that would and would not be appropriate to estimate with a Fermi calculation.
   2. Provide rough estimates for real-world quantities using "back-of-the-envelope" (Fermi) approximations.
   3. Evaluate the credibility of quantitative statements using "back-of-the-envelope" approximations.
   4. Use Fermi estimates to identify first, second, third order causes for example problems, and estimate their effect sizes.

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