The false positive (in fiction)

It’s not often that I find subtle math lessons hidden in works of fiction; much less blatant, straightforward, undisguised ones. They’re always welcome- especially when they reinforce something I’m already aware of.

An excerpt from Cory Doctorow’s Little Brother:

If you ever decide to do something as stupid as build an automatic
terrorism detector, here’s a math lesson you need to learn first. It’s
called “the paradox of the false positive,” and it’s a doozy.

Say you have a new disease, called Super-AIDS. Only one in a million
people gets Super-AIDS. You develop a test for Super-AIDS that’s 99
percent accurate. I mean, 99 percent of the time, it gives the correct
result — true if the subject is infected, and false if the subject is
healthy. You give the test to a million people.

One in a million people have Super-AIDS. One in a hundred people that
you test will generate a “false positive” — the test will say he has
Super-AIDS even though he doesn’t. That’s what “99 percent accurate”
means: one percent wrong.

What’s one percent of one million?

1,000,000/100 = 10,000

One in a million people has Super-AIDS. If you test a million random
people, you’ll probably only find one case of real Super-AIDS. But
your test won’t identify *one* person as having Super-AIDS. It will
identify *10,000* people as having it.

Your 99 percent accurate test will perform with 99.99 percent *inaccuracy*.

That’s the paradox of the false positive. When you try to find
something really rare, your test’s accuracy has to match the rarity of
the thing you’re looking for. If you’re trying to point at a single
pixel on your screen, a sharp pencil is a good pointer: the pencil-tip
is a lot smaller (more accurate) than the pixels. But a pencil-tip is
no good at pointing at a single *atom* in your screen. For that, you
need a pointer — a test — that’s one atom wide or less at the tip.

The “false positive cautionary” rears its head all too often; I’ve seen it on TED, in textbooks on probability, and even in a puzzle column in The Times of India- which is the last place you expect to find common sense.

‘Little Brother’ itself was very… didactic. This isn’t a review, so I’ll let a few more (spoiler-free) excerpts do the talking.

The law didn’t care if you were actually doing anything bad; they were
willing to put you under the microscope just for being statistically

I wasn’t the only one who got screwed up by
the histograms. There are lots of people who have abnormal traffic
patterns, abnormal usage patterns. Abnormal is so common, it’s
practically normal.

And finally, a few lines that really struck a chord, primarily because they’re intimately representative of a teenager’s awe.

If you’ve never programmed a computer, you
should. There’s nothing like it in the whole world. When you program a
computer, it does *exactly* what you tell it to do. It’s like
designing a machine — any machine, like a car, like a faucet, like a
gas-hinge for a door — using math and instructions. It’s awesome in
the truest sense: it can fill you with awe.

(Side note: Little brother is licenced under the Creative Commons licence and can be downloaded for free off the Internet.)


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