You are on holiday in some strange land, and you bump into Sue, an old friend. What are the odds?
Well, the probability is 1. 100%. It's certain. It bloody happened.
OK, you say, fair point - but that's not what you meant. What you meant is what is the probability you would bump into Sue, assuming you hadn't just bumped into him. I reply that that's a silly assumption to make as you just did bump into him, but you insist.
Well then, the probability is whatever you want it to be - pick a number, and I'll explain to you why it's plausible. You look perplexed and ask what I mean.
I explain: first of all, you need to specify the probability of what you are interested in.
-Do you want to know the probability you'd bump into Sue at the time and place you did
-Do you want the probability you would bump into an old friend while on holiday in general
-Or do you want the probability that something 'remarkable' enough would happen to you at some point in your life that would make you start asking silly questions about 'what is the probability of that happening'?
You say the first, obviously, and that I should stop being clever. I wasn't done, I say, and proceed to ask what is the information set I should base my probability estimate at - quantum mechanics aside, randomness is in the eye of the beholder after all, and if I was all-knowing God the probability of whatever it is that happened would be 1 even before it happened.
You throw your pina colada on my head and vow never to speak to me again.
A few days later, you are kind of missing me but don't feel like talking to me yet, so you visit bluematter. as a first step in rebuilding the relationship. And the first thing you see is this delightful little story, via Andrew Gelman:
In the city of Syracuse, the strangest thing happened in Tuesday's Democratic presidential primary.
Sen. Hillary Clinton and Sen. Barack Obama received the exact same number of votes, according to unofficial Board of Election results.
The odds of Clinton and Obama tying were less than one in 1 million, said Syracuse University mathematics Professor Hyune-Ju Kim.
Elaborating on Thursday, she [Professor Hyune-Ju Kim] noted: "The "almost impossible" odd is obtained when we assume the Syracuse voter distribution follows the New York state distribution. Since it is almost impossible to observe what we have observed, statistically we can conclude that Syracuse voter distribution is significantly different from the New York state distribution."
There would be less than one in 1 million chance of a tie occurring between Clinton and Obama in voting by a randomly selected group of 12,346 New York Democratic voters, she said.
To which Andrew replies:
Not to pick on some harried mathematics professor who'd probably rather be out proving theorems, but . . . of course Syracuse voters are not a randomly selected group of New Yorkers. You don't need a statistical test to see that. Regarding the probability of an exact tie: I don't think that's so low: a quick calculation might say that either Clinton or Obama could've received between, say, 5000 and 7000 votes, giving something like a 1/2000 chance of an exact tie. That's gotta be the right order of magnitude.
If there was one thing you were ever certain about, it is that you don't want to read what I have to say on this. A baseball bat happens to lie next to you (what are the odds!?). You grab it with both your shaky hands and smash the computer monitor to pieces.