For those who prefer video, this case study is described in the April 5th lecture of our Spring 2017 course.
This case study provides both a great example of how Fermi estimation can help you spot bull in the wild and of how bull need not be strictly false to be highly misleading.
Seventy million dollars! Wow, that's a lot of money. Sounds like a disastrously run program, right? Maybe even worth canceling, given that we're wasting government funds on scammers "including a state lawmakers and even a millionaire"?
Well, this is where Fermi estimation comes in handy. First of all, you may not know exactly how expansive the food stamp program is, but you could probably estimate that about 10% of Americans are on food stamps — or at least that it’s closer to 10% than to 1% or 100% (It's actually around 15%). Second, you're probably aware there are about 300 million people in the country. So around 30 million people are on food stamps. The actual number is about 45 million, but our estimate is plenty close for Fermi purposes.
If you are unfamiliar with the US food stamp program, you may not have a good idea of the average benefit paid to recipients annually. Still you could probably guess that it's closer to $1000 than to $100 or $10,000. (In fact, it's about $1500).
At this point, you've got enough information to see what's wrong with Fox's argument. Using your Fermi estimates, the US invests approximately
30,000,000 people * $1000/person = $30,000,000,000
in its food stamp program. That means that the fraction lost to fraud is
$70,000,000/$30,000,000,000 = 0.0023,
or around 0.2 percent. The actual fraction turns out to be less than 0.1 percent, but your Fermi estimate is plenty good to see what's going on. If there is any substantive inefficiency in the food stamp program, it certainly isn't fraud.
We don't know the average amount stolen by each fraudulent user, but even if SNAP frauds are receiving no more than the average legitimate recipient, fraudulent actors are a tiny fraction of SNAP benefit recipients. I'm assuming that Fox News is correctly reporting a $70 million dollar loss to food stamp fraud. But the notion that this might provide a motivation for canceling the program is utter bull. It would take an exceptionally punitive mindset to starve nine hundred and ninety-nine people in an effort to safeguard ourselves against one petty crook.
Update: Philip Bump at the Washington Post noticed the same flaw in the argument, and presents additional context in this post.
Another update: It looks like we were too generous in assuming that Fox News was accurate with its $70 million figure. The Department of Agriculture demanded a correction to the story, and Fox acquiesced. Amusingly, the $70 million figure that Fox used to argue that the foodstamp program is inefficient is far lower than the actual loss to fraud.
We can imagine the original writing procecss going something like this:
"Hey dude, what's a big, big number?"
"I dunno, how about [Dr. Evil voice] one million dollars?"
"No, even bigger."
"How about 77 million dollars?"
"Yeah, that's more like it! Thanks bro."