Fermi Estimations

I love this commercial… especially the way the owl says “two-hoo!” He also sounds like the owl on Winnie the Pooh.

The world may never know… if it weren’t for Enrico Fermi.

Fermi was a Nobel Prize winning nuclear physicist who worked on the Manhattan Project. One of his less significant (but more interesting) contributions was to the field of mathematical estimation.

Fermi showed that it was possible to accurately estimate the solution to an impossibly difficult problem by making a series of justified guesses. These impossibly difficult problems are often called “Fermi Problems”. An example of a Fermi problem might be the following: How many pizzas are eaten by BYU students every day?

I know, it’s tempting to just say “probably like a million” and move on with your life but this is solvable (and actually quite easy). Here’s how we might solve it.

  • How many BYU student’s are there? 32,000 (this isn’t exact, but it doesn’t have to be exact. It just has to be close, and I think it’s close)
  • How often does the average BYU student eat pizza? Once every 2 weeks (In other words, once every 14 days. So on any given day, a student has a 1/14 chance, or 7.14% chance, of eating pizza. We’re estimating anyways, so lets just round that to 7% to make it easier.)
  • How many slices does an average BYU student eat, when they eat pizza? 3 slices (some eat more and some eat less but the average has to be pretty close to this)
  • How many slices are in an average pizza? 8 (We know this one)

Now lets take our estimates and see if we can solve the problem:

Mathematical conversions, resulting in 840 pizzas a day

That’s a lot of pizza. And as you can see, it isn’t complicated math. It’s simple unit conversions.

The interesting thing is that Fermi showed that this methodology for estimating values is surprisingly accurate. Sometime your individual estimations are too high. Sometimes they are too low. But overall, it all evens out and you’d be surprised how close the outcome can be. I’ll bet my arm that if someone actually did the research, the number of pizzas we guessed would be within a couple hundred pizzas of the actual number.

(of course, I can make such a bold wager since I’m confident nobody will research this fact)

Lest you think that doing this kind of estimation is a pointless exercise, useful only for impressing the people sitting next to you at Comic-con, you should know that this technique once saved my bacon.

I was in an interview for a job I really wanted. Suddenly, the guy interviewing me smiled and said, “Ok, for the next question I’m going to throw you a curve-ball. How many golf balls can fit inside of a school bus?”

I smiled.

I then proceeded to describe how many golf balls could fit in a peanut butter jar, how many peanut butter jars fit on a standard bus seat, how many layers of these jars would stack to the ceiling, how many seats were on a bus, and how many jars would fit in the aisle. With some mental math (I did some serious rounding… otherwise doing it in my head would have been impossible) I got to a final number and threw in a few thousand extra balls for places I may have missed. Needless to say, I got the job.

Fermi estimations are sweet. Here are a few crazy problems that you can easily solve using Fermi’s techniques:

  • How many piano tuners are in Chicago? (this is the classic Fermi problem)
  • How many pencils would it take to draw a straight line along the entire Prime Meridian of the earth?
  • How many blades of grass on a football field.
  • What is the number of intelligent extraterrestrial civilizations in the universe? (this one has actually been calculated and the result is know as The Fermi Paradox)
  • How many notes are played on a given radio station in a given year?
  • How much milk is produced in the US each year?
  • If you drop a pumpkin from the top of a ten story building what is the farthest a single pumpkin seed can land from the point of impact?

…and you guessed it: “How many licks does it take to get to the center of a tootsie pop.”

Go ahead and see if you can figure it out (the actual answer is here).