Laymen Statistics Talk What does statistics has to say about this layman back and forth:

Layman A: The fact that John spilled his glass of wine on the table at that exact moment is peculiar. Never have I seen a man so
  masterful of his glass.
Layman B: Well, then, statistically, it is time that he had an accident.
Layman A: We are not talking about some random case. Historically, not once has he spilled anything. So statistically, it is extremely
  odd.
Layman B: Something must be wrong with your reasoning. It suggests that the first time of anything for anybody is extremely odd.

 A: There are several statistical issues that are relevant to this short dialog.

The fact that John spilled his glass of wine on the table at that exact moment is peculiar. Never have I seen a man so masterful of his glass.

One way to interpret this statement is: "I've spent a lot of time watching John using a glass and he's never spilled it, even though other people who I've spent the same amount of time watching have made several spills. So the spill is surprising." This makes sense: John's rate seems to be below average, so it's more surprising when the event happens in his case than in a typical person's case.
Another way to interpret it, particularly given the phrasing "at that exact moment is peculiar", is "It's surprising that he spilled his glass at this moment rather than an earlier or later one." This doesn't make a lot of sense, without anything special to distinguish this moment. If you randomly choose an integer from 1 to 1,000,000, and you get 280,782, then it's not peculiar you got this number, even though the chance was as small as 1 in 1,000,000. If you had earlier announced that you would get this specific number, that would make it peculiar.

Well, then, statistically, it is time that he had an accident.

This sounds like the gambler's fallacy: the belief that in a sequence of independent events, seeing one outcome repeatedly makes a different outcome more likely. If you flip a fair coin 100 times and get heads every time, your chance of getting heads on the 101st flip is still as high as $\tfrac{1}{2}$, even though the probability of getting 101 heads in a row is as low as $10^{-30}$.

Layman A: We are not talking about some random case. Historically, not once has he spilled anything. So statistically, it is extremely odd.
Layman B: Something must be wrong with your reasoning. It suggests that the first time of anything for anybody is extremely odd.

Here, A is invoking the fact that he's observed John a lot in the past. B seems to be ignoring these previous observations. If you see a man who you met today laugh, that's not odd. But if you've been friends with a man for 10 years and not once has he laughed and then, for the first time, he laughs, that's extremely odd.
A: May I offer yet another interpretation layer of this sentence:

The fact that John spilled his glass of wine on the table at that exact moment is peculiar.

Layman A places some special quality/value on this "exact moment". Even though we do not have specifics, we could assume that something interesting happened, apart from the spill. Maybe it was spilled on someone, maybe someone was announcing something and the spill caused a distraction, or even a small fire started and the spill extinguished the fire :) 
So it seems especially strange that the spill happened at this moment. It's "peculiar" as Layman A says. Implied in this is the notion that the spill may not have happened by chance. Maybe it was a deliberate spill, masquerading as an accident.
From a statistics point of view this is hypothesis testing. Given our observations and assumptions about prior probabilities, which of the two (or more) hypotheses is more likely? Layman A seems to be applying a "common sense" hypothesis testing, and tending to believe the not-an-accident hypothesis.  
