Enrico Fermi and the Oreo

How many piano tuners are there in Chicago?

How many quarters would you need to stack to be as tall as the Empire State Building?

If all digital data were stored on punch cards, how big would Google’s data warehouse be?

These questions are nearly impossible to answer accurately without extensive research.  Enrico Fermi, 1938 Nobel Laureate and nuclear power forefather, was known for accurately estimating similarly hard-to-know answers with next to no information and posing similar questions.  Most famously, he estimated the power of a nuclear blast by dropping bits of paper as the shockwave passed and then measuring how far they blew away.  His estimate of 10 kilotons was amazingly close to the official U.S. Department of Energy 21-kiloton yield determined 50 years later using gamma-ray spectroscopy, whatever that is.  His crude measurement led to a reasonable estimate within minutes.  The actual answer took decades to determine.

Incredible.

Using Fermi’s technique of applying known concepts and quantities, we too can develop strategies to estimate hard-to-know answers with little effort and decent accuracy.  In short, turn a random, nonsensical wild-ass guess into a scientific wild-ass guess or SWAG.


Munching on a chocolate creme Oreo the other day, I noticed a reversed wafer and wondered about Mondelēz’s quality control process.  I also pondered how many Oreos are manufactured in the U.S. every year.  For kicks, instead of Googling the answer right away, I opted to estimate it first, a la Fermi, and then see how I did.

First:  Population of the U.S.

  • I hear 300 and 330 million people get thrown around in the news.
  • 300 million is rounder, we’ll start there.

Second:  Number of households

  • Two people per household seems low. Four seems high.  Three sounds good.
  • 300 million people / 3 people per household = 100 million households

Third:  Households that consume Oreos

  • Oreos are popular but stores are stocked with all kinds of cookies.  Maybe ten percent?  Why not.
  • 100 million households * 10% = 10 million households consuming Oreos

Fourth:  Oreos consumed per household per year

  • My own empirical evidence suggests one bag of Oreos consumed per week
  • Each bag has ~40 cookies (varies based on standard size, Family Size, etc.)
  • 40 cookies/week x 52 weeks/year = 2,080
  • Call it 2,000 Oreos per household per year.

Fifth:  Final estimate

  • 10 million households * 2,000 cookies per household per year
  • 20 billion Oreos consumed per year.

That’s a lot of cookies.


How did I do?  Well, somewhere between horrific and abysmal.

But that’s ok!

It turns out, the U.S. Oreo production is not readily available, at least where I was looking. 

Mondelēz touts a worldwide Oreo production of over 40 billion per year and net revenue of $3.1 billion in 2019.  That’s about 7.8 cents per Oreo.

With the net revenue per cookie, we can estimate U.S. consumption with sales numbers:

  • 2016:  $742 million / $0.078 per cookie = 9.6 billion
  • 2017:  $674.2 million / $0.078 per cookie = 8.7 billion

Some sites took a direct route and just estimated consumption:

  • 1984:  6 billion per year
  • 2007:  205,000 bags per day or ~3 billion cookies per year
  • 2015:  778.8 million packages per year or ~31 billion cookies per year(!)
  • 2017:  7.5 billion per year

Combining these results, estimates range from 3 to 32 billion Oreos per year.  However, the 2007 and 2015 results seem to be oddballs compared to the other four.  Our range narrows considerably to 6 to 9.6 billion per year once we throw those out.  But now we’re stuck.  Without official U.S. production data, there is no great way to narrow the range further.

All said, I’ll go with 8 billion Oreos per year and call it a day.

Summary of Oreo production: Labels are year of estimate.
Included data   Excluded data My SWAG   Final estimate


Two things I found useful from this exploration:

  1. Billions vs. millions

I had no idea how Oreos Americans eat annually.  Before this, if someone asked me how many million are consumed annually, I would assume the number must be between 1 and 1000.  In the best case, I’m off by a factor of 10!  It turns out, my SWAG was bad, but not that bad.

For more extreme examples, try these questions out:

  • How many thousands of dollars is Amazon worth?
  • How many billions of people will read this blog post?

Messes with your head, doesn’t it?

Using “thousands” or “billions” frames the answer between 1 and 1000 when the right answer is either much smaller or much larger.  It is surprisingly persuasive, especially without prior knowledge.

  1. Time vs. accuracy

My SWAG took about two minutes to build.  The “close enough” result took about three hours of trawling through 10-K forms, press releases, and obscure blog posts to determine.

It took me 100 times longer to find a slightly more accurate answer.

For the sole purpose of satisfying a curiosity, that can be hard to justify.  Spending more time makes sense when accuracy is valuable, be it marketing or physics.

Well, that was fun!  Now you’ll have to excuse me.  I have a tall glass of milk and a stack of Double Stufs to attend to.


For more Fermi questions, check out these links:


Many thanks to Stew, Joel, Chris, Jesse, and Dan for your feedback. It really leveled-up this post!

Compound Writing is making me better.

Happy pi day!

Just wishing you all a happy pi day.  3/14/16 turned into decimal form (3.1416) rounds pi nicely to the nearest ten-thousandth.

This is much catchier than wishing everyone a happy “Ides of March eve”, which just sounds dreadful.

www.pidaychallenge.com/ makes for some entertaining mathematical puzzling.

www.expii.com is worth a look too with various problem solving sets in math and science set up by Carnegie Mellon math professor Po-Shen Loh, National Coach of the USA International Math Olympiad team.

Georgia Tech hosts a pi mile road race, which has run since 1975, and is actually a 5k (3.1068 miles, not 3.14159265… miles) since a change in 2002.  This is, confusingly, not run on pi day, but toward the end of April.  A portion of the course is on the Tyler Brown Pi-Mile Trail on campus dedicated to a former student government president and military serviceman killed in action in Iraq in 2004.

Enjoy the festivities, wherever you may be!

The leap day

I couldn’t resist posting on February 29th a little piece I wrote, well, about four years ago…


 

February 29th has come and gone with its usual fanfare being celebrated quadrennially.  That is, except every century when we skip the leap day, except every four centuries when we keep it.

Huh?  Perhaps I should back up a bit…

Our typical calendar year is 365 days (or 8760 hours, 525,600 minutes, 31,536,000 seconds, you get the point…) but our actual solar year, i.e. the time it takes the Earth to make one rotation around the sun, is approximately 365.242190419 days.  If we were to keep using 365 days per year, after a century December 31st would act a lot like a typical December 7th because we would be 24 days behind.  After 1000 years, we would be off by 242 days and celebrating New Year’s Day at a time when spring was just getting into full swing.  This is clearly a problem!  Hence, the “Leap Day” of February 29th.  Adding this every four years and our average calendar year becomes 365.25 days.  A much better result, but clearly we can do better!  To trim that average to 364.24 days, we skip the Leap Day every one hundred years.  So why, you may ask, did we have a leap day in 2000?  We celebrated this quadricentennial event because an average of 365.24 days is just not good enough!  By adding that leap day back every 400 years, our average calendar year becomes 365.2425 days.  This amounts to a deviation of 26.7 seconds per year or 0.0000848% and a much better result.  Of course, after 3200 years of this, we would be off by an entire day again, but that debate can wait a couple millennia.

What are the odds?

$1.3 billion!  That’s quite a hefty jackpot for the next Powerball drawing, even if the number is deliberately inflated to raise the hype.  Still, the lump sum payout is estimated to be $806 million; not a bad day at the office.

The odds are posted around the web, but how are they determined?

The updated version of Powerball started in October; five numbers are drawn from a set of 69 white balls and one number from the set of 26 Power balls.  To win the jackpot, your five numbers and one Powerball number have to match up with the drawn numbers.  Your chance of one of your five numbers matching the first ball pulled is 5 out of 69 (7.25% probability).  Once a ball is pulled, it is not replaced, hence the chance of one of your four remaining numbers matching is 4 out of 68 (5.88%).  This continues until the five white balls are chosen (3 out of 67, 2 out of 66, and 1 out of 65) and then the Powerball at 1 out of 26.  When all is said and done, out of the 35,064,160,560 ways the six total numbers could be pulled, you could win 120 different ways, since the order that the numbers are pulled doesn’t matter (permutations vs. combinations anyone?).

odds
Odds are 292,201,338 to 1

When does it makes sense to play?  Well, Walter Hickey has made a name for himself looking at this sort of thing.  His 2013 article says it only makes sense if you take the annuity and if the jackpot is greater than $345 to $380 million.  Mind you, this was when Powerball was way easier to win (odds were a paltry 175,223,516 to 1 back then!)  With the tougher odds, breakeven, assuming just one winner, looks like this:

expected value

Once the jackpot hits $491 million, you can statistically justify purchasing a ticket, since the expected value is above $0.  Of course, that is **AFTER TAXES**.  With a top federal tax bracket of 39.6% and state taxes ranging from 0% to 8.8% and some cities adding their own levies as well.  That means it only works with jackpots from $813 million to $1.029 billion!  Assuming multiple winners, which is more and more likely with more tickets purchased, that minimum number keeps going up.  This is getting ugly…

Long story slightly longer, the odds are such that it is nearly impossible to statistically justify purchasing a ticket.  Am I playing?  Uh… yes, I am.  Why?  Consider it a cheap insurance policy.  Should the office pool hit, I’ll have options.  Besides, it’s entertaining to share in the thrill of anticipation, even if it will undoubtedly end with a tiny payback.