8.2 Fermi Problems

Estimating quantities based on what we know.

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.

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.

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).

Useful Resources

Lecture VideoDiscussion SlidesHandouts and ActivitiesReadings and Assignments

Discussion Slides Template

The discussion slides for this lesson.

Fermi Problems Worksheet

Worksheet used for all the activities in this lesson.

Solved Fermi Practice Problems

A PDF with solutions to some trickier Fermi problems.

How Many Licks: Or, How to Estimate Damn Near Anything

Aaron Santos on Fermi estimates.

Recommended Outline

Before Class

Familiarize yourself with the worksheet and print it.

During Class

5 Minutes	Introduce the lesson and go over the plan for the day. Make sure people have groups, spokespeople, etc.
15 Minutes	Run through the gasoline spending activity.
45 Minutes	Let the students work in small groups on the government spending activity.
15 Minutes	Use any remaining time to work on the optional additional problems. If you don't have a separate lecture to go through the solutions to the government spending problems, you should use this time to do so.

Lesson Content

American Spending on Gasoline

In this activity, you break down an especially complex Fermi problem into its constituent parts in front of the class. This demonstrates to these problems get broken down.

Students sometimes read too much into the complexity of this example. They over-complicate the problems they solve on their worksheets. Make sure to emphasize the importance of breaking down problems into the quantities that are easiest to estimate and aiming for first order causes. Going into further detail is often unnecessary and may not even improve the quality of our estimates.

Instructions

1 Minute	Quickly poll the students on how much they think Americans spend on gas each year.
14 Minutes	Break down the actual problem as shown below. Explain how the units cancel out. You may draw this tree diagram step by step on a whiteboard. As you go through the thought process, have the students shout out their estimates for each of the quantities listed.

Quick Poll

Ask the students to quickly guess answers to the question, "How much do Americans spend on gas each year?"

1. Between zero and 10 million?
2. Between 10 million and 1 billion?
3. Between 1 billion and 100 billion?
4. Between 100 billion and 1 trillion?

Rather than just giving your students the answer, you demonstrate how it can be solved with a Fermi estimate.

Estimation Steps

You can break down the estimation as per the figure.

How close was your students' original order-of-magnitude guess to our final Fermi estimate?

Optionally, ask your students how would we go about calculating the plausible upper bound and lower bound estimates? For what purposes might it be useful to calculate upper and lower bounds?

Estimate an upper bound and lower bound for each estimate entering the Fermi calculation, where a rough number isn't known. Calculate it out to get a high bound and low bound.

This might be useful if you're particularly unsure about some of your estimates.

Government Spending

In 8.1 Orders of Understanding, students simply guessed the order of government spending in three categories. In this activity, students will work in small groups, using Fermi estimation to get more concrete estimates for these quantities.

These problems should not take as many steps or components as the gasoline example.

Fermi Problems Worksheet

Worksheet used for all the activities in this lesson.

Instructions

2 Minutes	Share the worksheet with your students.
43 Minutes	Let students work on the worksheet in small groups and frequently offer assistance and guidance, without giving away any answers. Feel free to move on to the optional Fermi problems below if students finish this activity early.

Additional Problems

This activity is optional and should only be done if you don't have the time to demonstrate solutions to the government spending problems in a separate lesson.

Have the students estimate in small groups any of the following quantities. Encourage them to give a lower and upper bound, as opposed to a single value. These problems were chosen as as to have a diversity of units and types of quantities being estimated.

1. Amount of food thrown in landfills in America every year.
2. The weight of all food an army battalion would have to bring on a 200-mile march across a rainforest.
3. The amount of water used to irrigate all the front lawns in Los Angeles in one year.
4. The total weight of tea (leaves) consumed by the British in one year.
5. As a laptop keyboard engineer, how many repeated key presses would you have to rate any individual key for, so that consumers do not typically encounter the malfunctioning of any key?
6. How much water would hotels in America save if they all went from refreshing guest towels daily to refreshing them only when left on the floor or for new guests?
7. How much clothing gets thrown out in the United States?
8. How many ties were bought in the U.S. in 2019?
9. How many pieces of paper does the average American college student go through in four years?
10. What percentage of the US GDP changes hands on an average Wednesday?
11. How much money, in total, did Americans spend on restaurants in 2019? (these 2019 questions are because Covid mixed things up.)
12. What percentage of total adult life hours is spent taking care of children? (Restrict to the US to make easier.)

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