Mark Chu-Carroll over at Good Math, Bad Math gives a good simple description of how the financial markets got themselves into this mess.
He also got me thinking about a particular problem that I’ve found jiggering around in the back of my head during my last few forays into the Interwebs: people really don’t understand the Strong Law of Large Numbers, and the immediate consequence of this is that they don’t understand that really improbable events are usually impossible. Not unlikely. IMPOSSIBLE.
Let me illustrate. Go find a pair of six sided dice. Give ’em a good shake, like you’re in Vegas, and give ’em a toss (you’re encouraged to call out, “Daddy needs a new pair of shoes!”). Now, without me knowing what the result is, I’ll go ahead and guess “7”, as it’s the most likely result… but of course there’s a substantial probability that I’m wrong. The actual probability that you rolled a seven is 0.166666666(…)
Now grab a pencil and paper, and write that number down. Roll the dice 10 more times, and record all of your results. Add them together and divide by 10, scribble that in another column. Now roll the dice 10 more times, and add the 20 results together, and divide by 20… scribble that in that other column. Repeat until you’ve rolled the dice 100 times.
Now you can look over that sheet of 100 dice rolls, and you may very well find some “crazy improbable miracle result!”, like rolling 10 twos in a row… but you’ll find that that second column of numbers averages very close to 7, don’t it?
What the Strong Law of Large Numbers says is that if you roll those dice over and over and over again, all the way out to infinity… the probability that the average of those dice equals seven is 1.
It’s not close to 1, it is 1. Certainty. There is no other possibility.
“But Pat,” you may think, “I could keep rolling two the hard way! It’s possible! Each die roll is an independent event! ANYTHING is possible!”
You’re right, each die roll is an independent event… but your conclusion, that therefore “ANYTHING is possible”… no, it’s not. Taking individual events, anything is possible (I could always roll a “2” on my next die roll)… but while each die roll is an independent event, our average is NOT. It depends upon all of the other results that we’ve rolled so far… so while each die roll is independent, the average is conditional (note – misunderstanding the converse of this and all of the implications of it is where the Gambler’s Fallacy comes from… some people understand intuitively that the average is conditional, but then assume that this has some “effect” on the individual events).
Now, there’s a lot of convergence theorems in mathematics as well, which predict how quickly your average is going to get arbitrarily close to the expectation, but we don’t want to get toooo complicated here.
Basically, what the math shows is that an outcome requiring an infinitely large collection of improbable events to occur is not improbable… it’s impossible. Taken with convergence theorems, it’s actually provable that given a sufficiently large collection of improbable events, we can conclude that the outcome is impossible. Surprisingly to many people… “sufficiently large” isn’t that big at all, particularly when the events are really improbable… particularly when you take into account boundaries.
This is why followers of Popper and Hume are wrong. This is why, even though science is not an axiomatic system, scientists have a tendency use the terms “proof”, “law”, and “truth”… in a purist sense, they shouldn’t, but in an empirical sense, it’s certainly reasonable for them to do so.