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.

Relation to Other Lessons

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 GoalsDefinitions

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).

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