(Secretly, I’ll be looking for the nearest exit. But, oh yes, I’ll laugh.)
Whenever any weird, comical, or seemingly unusual event happens—like when I was walking with a friend and a school bus drove by and a kid threw a crayon out the window which landed plop into his pop1—and a guy like me is around, somebody hilariously asks, “What are the chances of that!”
Here’s the universal answer: if it happened, then the chance is 100%. I’m not kidding. That’s the real answer. Why? Because all chance is calculated conditional on circumstance. And the circumstances were whatever it was that caused the strange thing to happen. And since the strange thing happened (as it was caused to) the chance is 100%.
If that sounds like a lawyer’s answer that’s because nobody ever means the question in this sense. They want to know what are the chances the strange thing happens again.
Same answer: 100%. It’s the same answer for the same reason. Because all chance is calculated conditional on circumstance, and the question does not specify any change in circumstance, the probability must also remain unperturbed.
Probability isn’t alive, it is a zombie, a perfect slave to the information which specifies it. If nothing changed, if the world really were to reset itself to the precise same condition before the strange thing happened, and there is no change to the physical laws and (all) causes which drive everything, then the strange thing must happen again.
But nobody means the question in this sense either.
So what sense do they mean it? That’s not clear. Two things are needed in any chance or probability calculation. The first is the proposition of interest, which is our strange thing. When somebody asks “What are the chances?” they clearly mean to keep this proposition intact. The second thing, as discussed, are the conditions. We’ve already seen that we’re not meant to keep these conditions precisely identical, so people must be imagining these undergo some sort of shift. What kind?
Let’s switch examples in the hopes of clarification: an American roulette wheel. The proposition is “The ball lands on the 5”, which isn’t especially strange, but the proposition isn’t the problem. What are the circumstances? A bunch of slots (38) on a wheel, one of which labeled “5”, and a ball which must land in one of the slots. Probably, based on this and no other evidence (“no other” means no other), is 1 in 38.
Now if the ball in fact landed on 5, and somebody asks, “What are the chances of that happening again?” we’d say “One in thirty-eight”. Why? Not because we expect the circumstances to be identical on the next spin—because if they were then a 5 must again show—but because we keep the same evidence in mind (that the wheel has 38 slots, etc.).
Inveterate gamblers start the same but change the evidence. They say to themselves something like, “It just came up 5, it probably won’t do it again. Of course, the wheel still has 38 slots and so forth.” Now the probably the inveterate gambler then calculates based on this changed evidence probably won’t be zero, but he’d say it was close.
And that’s okay, because he did change the evidence (as I cannot emphasize enough). A pedant would wag his finger at the gambler and insist he’s using the wrong evidence. Well, and so he is, but the gambler’s not coming to the wrong probability based on the evidence he used.
The best evidence to use nobody has: that would be the full physics of the circumstance (weight of wheel, forces on the ball, friction of the slots as ball hits, and on and on). Now this sounds a bit like listing the circumstance for the strange thing, doesn’t it? But if we had that information, we’d know whether the 5 was going to show with certainty.
So where do we stand? For the strange thing, the asker means to change the circumstance/evidence, as we saw, but he never specifies how to change it. We know the circumstances must be different to get a probability different than 1, we just don’t know in what direction they should be modified.
Since we can’t know what aspects of the circumstances should change, there’s no way to know what the probability of the strange thing is. Consider the crayon. Should the bus be moving or stationary? At what distance is the cup? What color crayon? Should color matter? Which angle is the cup mouth facing the bus? Is the cup moving or stationary? And on and on endlessly. Unless we are prepared to say exactly which evidence we’re considering, there’s no way to know the chance.
What a disappointing answer.
1That’s right. I said pop. You barbarian.