If I am reading your question right, you are asking me if flipping 1000 heads in a row is synchronicity. No, it isn't. Synchronicity isn't about how far the odds are or what the event is, it's about an emotional response to a series of coincidences.
Rodney disagrees with you. He insists that the low probability of an event is in part what makes it "synchronicity".
And in fact many people who believe in synchronicity (or being in synch with the universe, as you phrased it, which isn't far off from The Secret nonsense) will support their case by asking, "What are the odds of meeting someone I know when I'm out of the country?" or whatever the event is. But they don't listen when it is explained to them that the odds aren't that long when you consider how many possible chances there are for an arbitrarily defined "amazing" event.
I understand that you aren't making the Texas Sharpshooter fallacy, but Rodney most certainly is.
All this talk about the odds of 1000 heads in a row have no bearing on the emotional state of the observer. Discussions about syncronicity should lean more toward psychology, not probability.
ETA, I think we are actually agreeing. Pareidolia is a human construct, seeing a pattern where none exists, discussing the odds of a face actually appearing doesn't really play into it, it's a psychological effect, and such is sychronicity. The event is insignificant, it's the person's emotional response that is significant.
So what's the point of denying that it's a mere coincidence and insisting that it's an example of synchronicity? (That's how the term is used. It implies an inherent meaningfulness in the event because it's a low probability event. Then you have people like Rodney claiming that because it's low probability, it's impossible that it could happen by chance.)
I partially agree with your statement, "Discussions about syncronicity should lean more toward psychology, not probability." I think it's better explained, as I have repeatedly, but the fact that we evolved as intelligent animals living in very complex social groups. The mental capacity and tendency to see patterns, to infer intention, etc. has a distinct adaptive advantage. It comes at the costs of the tendency to make Type I errors. I don't know if you consider that to be a "psychological" explanation; I'd prefer to call it the biological explanation--or perhaps the natural selection explanation.
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I think Rodney's problem (and I apologize in advance for putting words in his mouth) is that he doesn't believe that the 1,000-heads-in-a-row result is just as likely, or unlikely, as any other. Intuitively, it doesn't "feel" right, as I can attest, being a mathematically challenged person. When people explain odds and so forth to me, I can understand it, but it still "feels" like a pattern like 1,000 heads in a row should be less likely than a more "random" result. Of course, the fact is that until a coin toss happens, any potential result is just as random as any other.
How about the following (something I described earlier in this thread), which might be more accessible without knowing the maths.
Imagine dealing out Ten, Jack, Queen, King and Ace of clubs from a well shuffled (randomized) regular deck of cards. The odds of getting that hand are less than 1 in 2.5 million.
Now imagine dealing out 3 of clubs, J of diamonds, 5 of clubs, 3 of hears and 7 of spades. The odds of dealing out that exact hand are also less than 1 in 2.5 million.
So clearly, the improbability of an event is not what makes it significant and meaningful. It's something about the
perceived pattern.
Or better yet, consider Fromdownunder's example: think how improbable it was for all the events from his conception until a couple of days ago that led up to his posting on this forum. The odds are staggering! Yet few people claim such a mundane event is "synchronicity". Why?
I say it's because synchronicity is just another Type I error--it's based on seeing a pattern or intention (even if the "intention" is just something impersonal like "karma" or "being in tune with the vibrations of the universe" or whatever). I've explained why the human mind does this--basically it's the cost of avoiding Type II errors.