Ivan Seeking said:
There is no need for language trivia or discussion of the trivial case. We are talking about events that are perceived to be statistically unlikely. The trouble is that we have to define what statistical relationship exists. And then determine how often one would expect a 1:100, or 1:1,000,000 event, or whatever the odds are, to occur.
I brought this up before a few years back, but I guess it bears repeating. Feynman deftly pointed out the irrelevancy of statistical probability to any given event:
"You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance I would see that particular one tonight? Amazing!"
Any given
specific event is statistically
extremely improbable. The more specifically you define the event the more true that becomes (that
particular license plate on that
particular night!). As Georgina pointed out, we only notice the incredible improbability when the specific event has personal significance. The odds of running into a person you haven't seen in 30 years at the store one day are actually about the same as the odds of running into a specific individual you've never encountered before, if only you appreciate how specific that specific individual actually is, and how specific that time and place. Each stranger, each time, each place, is a very specific. Once you pay attention to that, and focus on how specific they are, the odds of you encountering them become less and less probable. We beat unbelievable odds moment by moment, all day long.
As Georgina emphasized, the key word is "perceived". We are pattern-seeking creatures, with a distinct leaning toward giving everything a kind of "pattern test". Very small whiffs of familiarity put us on alert and we test them to see if they fit a pattern we know. By this mechanism, a circle, two dots and an arc are "recognized" as a smiling face:

, when in fact it bears no authentic resemblance to any face in nature. We even accept it rotated 90 degrees, without the circle :) It's a stripped down abstraction that never-the-less works due to our propensity for checking for patterns at many different levels of perception. Certain kinds of specificity take on extraordinary importance. Other kinds, though equally specific, are ignored, discounted.
Calulating the probability of an event ends up being immaterial in determining if it was a coincidence or not. If you define the event according to certain parameters it become statistically impossible that it should ever occur. Define it according to other parameters, and it becomes inevitable that it should occur.