8.2 Fermi Problems

Estimating quantities based on what we know.

Useful Links

• Worksheet
• Three Column Overview of the Week
• Lesson Slides (2023 Master)
• Website Page

Readings and Assignments

• How Many Licks?: Or, How to Estimate Damn Near Anything>

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.

Definitions

• 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:

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

Tutorial Video

Nikolai's tutorial to share with students:

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.

Before

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.

After

14.1 Scenario Planning

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

Recommended Outline

Before Class

• Review PlayPosit and discussion questions and ask faculty, Gabriel, or Emlen any questions you have.

During Class

• (5 min) Come up with some fun way to assign the roles of spokesperson and notetaker (e.g. earliest birthday in the year, lives furthest from campus). Remind them of the responsibilities of these roles.
• (20 min) Run through the gasoline spending activity.
• (45 min) Let the students work in small groups on the government spending activity.
• (5 min) Use any remaining time to work on the optional additional problems.
• (5 min) Collect questions for plenary.

Lesson Content

American Spending on Gasoline

In this activity, you break down a complex Fermi problem into its constituent parts on the whiteboard so that the students get to see how these get broken down. See minute 22 of this video from a previous year to see a demonstration.

Instructions

1. (30 seconds) 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?
2. (10 min) 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.

Estimation Steps

You can break down the estimation in the following way.

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

Optional: 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?

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.

Instructions

1. Share the worksheet with students.
2. (No more than 45 min) 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.
3. Reassure the students that Saul will walk the whole class through these estimations in the plenary session.

Additional Problems

(Optional given enough time) 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.

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. The total amount of labour-hours needed to grade all the homework, projects, quizzes, and the final for L&S 22 in one semester. (How many 10 hrs/wk undergraduate readers do we have to hire to complete this task?)
6. 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?
7. 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?
8. How much clothing gets thrown out in the United States?
9. How many ties were bought in the U.S. in 2019?
10. How many pieces of paper does the average Berkeley college student go through in four years?
11. What percentage of the US GDP changes hands on an average Wednesday?
12. How much money, in total, did Americans spend on restaurants in 2019? (these 2019 questions are because Covid mixed things up.)
13. What percentage of total adult life hours is spent taking care of children? (Restrict to the US to make easier.)

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8.1 Orders of Understanding

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9.1 Heuristics