Wednesday, June 24, 2009

probabilistic intuition

It often surprises me how badly humans naturally think about math, and specifically probability. Many of these instinctive mistakes are so frequently applicable to everyday life that I can't fathom how our brains did not evolve to immediately see them.

For example, basic logic. If A implies B, it's not true that not-A implies not-B. It's also not true that B implies A. Yet people more frequently make these false claims than they claim that not-B implies not-A, which IS true. (For example: rain implies wet streets. Wet streets do not imply rain. A lack of rain does not mean the streets are dry. But if the streets are dry, it definitely isn't raining.) Luckily, I've found that habitual analytical exercise (ie lots of math proofs) very quickly eliminates logical mistakes. Or at least, makes the ones you instinctively make much more arcane than contraposition.

Then there's basic probability. Easy statements about probabilities are in fact so unintuitive that even with years and years of practice and experience with math, I constantly have to formally check things in my head when it just sounds wrong. Probably the most common mistake involves adding independent probabilities. Say every time you get on a motorcycle you have a 1% chance of dying. It is NOT true that if you get on a motorcycle twice, you have a 2% chance of dying. It's like flipping a coin until you get a heads. Each time you flip, you have a 50% chance of succeeding. But doing it twice does not mean you have a 100% chance of succeeding. The expected number of heads you will flip in total does double, but not the chance of succeeding at all. This is probably mind-numbingly obvious to anyone who read this, but if you start paying attention to TV scripts, books, casual dialogue of any kind, the mistake is astoundingly ubiquitous. (Noticing them, and giggling at them, is one of my favorite nerdy pet pleasures. [Venturing into double parentheses territory, is pet pleasure actually an antonym for pet peeve? I declare it so.])

Of course, if we can't even intuitively add probabilities correctly, we have no chance at all when Bayes law is involved. If there's a 1 in 10,000 chance I have AIDS, and a 1 in 100 chance of a false positive on an AIDS test, you can be darn sure I'll be terrified if I test positive, even though that only gives me a 1 in 100 chance of actually having AIDS.

Another mistake I always found a little comical involves ignoring information that the answer depends on. Back in junior high when I was a big Carl Sagan fan, I always cringed when hearing believers in extraterrestrial life make the following argument: "If life is so rare as to only occur once in the universe, what are the chances we would end up on that exact planet? Clearly, life must be more common." While I agree that life is probably much more common than once per universe, even if it is that rare, there is no other planet we could possibly have ended up on than the right one, by the mere fact that we ARE alive!

Of course, more severe limitations to our probabilistic intuitions occur not with these examples that apply every day, but to situations that we, for obvious reasons, did not evolve to be able to handle mentally. For example, the scale of the universe, both on the large and small ends, and the timescales involves to create it. I'm reading a fantastic book by Richard Dawkins right now (The Blind Watchmaker) and he does well to explain our view of probability as a subset of an entire spectrum of likely events. Since we are only alive for less than a century, things that only happen once every million years seem completely miraculous to us. Not so to someone who lives to be a billion. It's no wonder young-earth creationists think us scientists are off our rockers. 6000 years is about all I can fit in my brain comfortably, too.

It's a wonderful thing about science that we can step outside of our human limitations (maybe not all of them, but many) and make objective observations of the universe. It may seem insane to our feeble minds to propose that a stone archway formed over thousands of years of slow water erosion. But as scientists, we can carefully study the process and calculate its actual probability. Whoever said that science kills the sense of beauty and mystery in nature had it completely backwards: only science reveals the astonishing nature of the universe we're part of.