(In the original version of this post, there were two places where I wrote “cubic” instead of “square”. It was a stupid mistake, but just a stupid wording mistake. The calculations are unchanged: I was calculating square units, not cubic – I just typed the wrong thing.)

I haven’t written about this in a while, but for a long time, I’ve been fascinated by just how bad humans are at actually understanding big numbers. When it comes down to it, we’re comfortable with numbers we can count on our fingers. The further you get from our fingers, the worse we get at it.

We see this over and over. I’ve been noticing a new variation on this lately, in discussions about the missing Malaysian airliner. For example, I was at the dentist tuesday morning for a tmj mouth guard because I’ve been told that I grind my teeth at night. They’ve installed TVs in the exam rooms, to give you something to do while you wait for the novocaine to kick in, and they had it turned to CNN. The technician who was setting things up was talking to me about what they were saying, and she kept repeating: it’s a plane! It’s huge! Why can’t they find it?

If you think intuitively, it’s a good question!

In fact, if you look at it with basic intuition, it seems insane. A Boeing 777 is a huge plane! The thing is nearly as long as football field, and it’s length wingtip to wingtip is just short of 200 feet. Fully loaded with fuel at takeoff, it weighs more than 300 tons! How can you lose something that large? Even if it broke into pieces, the debris field from it would need to be huge, and it must be easy to find!

But that’s relying on intuition, and as I said in the introduction, our intuitions scale very poorly. To show what I mean, I’m going to work through a process that will hopefully help make it clear just how badly our intuition fails us here.

We’ll start by coming up with an estimate of the quantity of debris.

Assume that the fuselage is a continuous tube. The official stats say that the diameter of the fuselage is 20 feet, and it’s 242 feet long. So a deliberately overlarge estimate of the total surface area of the fuselage is 24220π – or about 15,000 square feet. Assume that the surface of the wings is about he same, and you get a total surface area of about 30,000 square feet. That means that if the entire body of the plane was peeled and laid flat, and it all floated on the water, it would cover 30,000 square feet of the surface. (In reality, much of the plane would sink, but the seats would float; on the whole, it’s probably a wash; the 30,000 square feet is probably still order-of-magnitude correct for the amount of debris visible on the surface.) Sounds like a whole lot, right?

To get an idea of how easy that is to find, we need to consider how much area we need to search. From what we know about how long the plane could have been in the air, and what direction it could have been moving, we’ve got a search area that’s at least a couple of thousand square miles.

One square mile is 27878400 square feet. So assuming that the plane crashed, and its wreckage is distributed over one square mile, that would mean that in the crash area, if every bit of the plate floated, just over one tenth of one percent of that square mile is covered in wreckage. That’s piling together very unrealistic assumptions – a highly constrained debris field, all of it floating – in order to stack the odds of finding it!

We’re using some optimistic assumptions here – but even so, even with a debris field that’s highly (unrealistically) constrained, it’s still very sparse. (In fact, it’s likely that if the plane crashed in the water, the debris field is spread over more than one square mile.) That’s what we’re looking for: an area where less than one tenth of one percent of the surface is covered, in an area covering several thousand square miles.

A thousand square miles is pretty squarely in the zone of stuff that we don’t really understand very well. Getting an understanding of just how big an area a thousand square miles is… it’s not something that our intuition will help with. But statistically, we’re looking for a patch of debris where the visible artifacts cover less than one ten-millions of one percent of the area being searched. If the search area were a football field, you’d be looking for a little fleck of tinfoil, 1/32nd of an inch on each side.

Only even that is making it too easy: it’s not one piece of tinfoil, 1/32nd of an inch on a side. It’s a fleck of foil which was shredded into a thousand even tinier pieces.

That’s assuming a search area of only one thousand square miles. But the search area is, potentially, quite a lot larger than that. Searching for the tinfoil on a football field is easy in comparison.

Once you understand how small an airplane is in comparison to a small patch of ocean, and you understand just how big a small patch of ocean is, then it’s not at all surprising that we haven’t found it yet.