Math geeks: odds question

I got an IM last night on AIM from an apparently random person; at least this person claimed to have found me completely at random. This person, unlike many random IMers, used complete sentences with grammar, spelling, punctuation, all the bells and whistles. Now, that in itself is unusual enough that I decided to talk to him (I’m assuming the person was male, mostly because when asked to identify himself, he said “Just some guy from Boston.”). This was actually the second time I’ve gotten a literate random IMer, and I regretted not talking to the first one, just because that’s so rare.

I asked him if he had found me through the SDMB or completely at random, and he said random, but he knew of this board, and in fact used to lurk here until 9/11 stuff drove him away. I asked again if he really had found me at random and he insisted he had. In conversation with mrblue, he pointed out that it’s no more bizarre than his discovery that his next-door neighbor posts here.

I still wonder, though, what the odds are of such a thing happening. Out of the millions of AIM users, what are the chances that two random people will both be members of the SDMB? Particularly since this one and the other literate one were the only random IMs I think I’ve ever gotten.

Well, suppose N is the number of AIM users, and n is the number of AIM users who are also Straight Dopers.

Then there are n(n-1)/2 possible pairs of “AIM users who are also Straight Dopers” and N(N-1)/2 pairs of AIM users.

So the odds that a random pair of AIM users are also both Dopers is the ratio of those two numbers: (n[sup]2[/sup]-n)/(N[sup]2[/sup]-N).

Now just fill in your best estimates for N and n.

Yeah, I figured it would be something ridiculously easy like that. The rain here is making my brain foggy, that’s my excuse. Now would the odds be substantially different if we try to figure in it being the first random IM ever?

If you mean “is it more likely that at least one of my first, say, 20 IMs is from a Doper than it is that my first IM is from a Doper”, then yes. If you mean “Is my first IM less likely to be from a Doper than, say, my 20th IM” then the answer is no, barring any other information; the odds for your 1st and 20th IM, individually, are the same.

The problem, though, is that if this guy’s telling the truth, it’s too late for you to ask this question. The odds are one.

We’re talking about a theoretical random IMer, since the guy who IMed me is now no longer random.

I think the question was “what the odds are of such a thing happening,” so I’m not so sure.

On the other hand, there is a bit of selection going on–you were focussing on literate IMers. Of course, literate IMers are more likely to be dopers. Sheesh.

I wasn’t focusing on them, those two are seriously the only two random IMs I’ve ever gotten. I keep seeing people post about idiots they’ve been IMed by, so I’ve gotten the impression that that is more the norm–both of these guys were not only literate, but seemed to actually want to chat rather than cyber. Odd.

One thing to keep in mind is that the odds of a particular odd thing happening are way smaller than the odds of any odd thing happening.

For example, the odds that I’d get exactly my license plate number are something like 1 in 100 million, but the odds that I’d get some license plate were 100%.

As Geobabe, I’m sure you already know this - I’m just using this as an opportunity to point out to others how coincidence is a certainty. Eventually, something weird will happen to you, so don’t think it’s aliens or crop circle gremlins or anything.

Well, is there a directory for AIM names? How does one select a destination for a random AIM encounter? Do you have a profile somewhere?

I’m assuming, given that you get almost no random IM’s, that you don’t have a profile somewhere stating your gender. According to your SD Profile, your IM name is rocksci. That would tend to select for the geologically inclined. And a reasonably intelligent person on the internet who is bored enough to do random chats with people is likely to have had enough time to have done enough surfing to have run into the Straight Dope MB.
But you can run an experiment – change your IM name to GeoBabe and see how many random IM’s you get.

The odds will be gooder, but the goods will be odder.