That was not the argument. "Outside the realm of chance" in your case is connected with a single person that can repeat the performance, but the argument was about testing lots and lots of persons, and then you will eventually get a positive result. The chance that the performance can be repeated is of course slim.
It depends on how positive the result is, which goes back to my original point.
Let's say we've tested, oh, ten thousand people over the years in various controlled psychic tests. The results would form a bell curve with some people performing horribly, most people being average, and some who've done really well. Two points to make:
1. For the people who do really well, how far do we push the chance hypothesis? A 1-in-a-100,000 result? 1 in a million result? 1 in ten million? Different people are going to point to different results as significant. Maybe for a scientist it's one-in-a-billion. Maybe for me, who's on the fence, it's one-in-a-million. Maybe for Suzie, who's into New Age stuff, it's 1 in 100,000. How do we decide who's right? We're going to fall back on our worldviews, and the discussion will get very philosophical and won't produce any results.
2. The number of people we test is only significant if we assume psychic ability is uniformly distributed across the population. That the chance of displaying some psychic ability is like the chance of winning the lottery: equal for everyone. But that's just an assumption. What if only 5 in 10,000 people have any ability at all? What if I test 10,000 people, and five score really high (they beat the odds by 1 in 100,000). How do I know I haven't just identified the five people who would have scored high had I just tested them? I would retest them, over and over again, and see if there's a regression to the mean, but obviously a lot of people aren't going to want to be tested over and over again. So if I do a large trial, and get some anomalous results, and those people don't want to be in the follow-up trial, how should I interpret the results? It's going to come down to how one views reality, and while you might not agree with the theist, or the immaterialist, or the person who thinks we're living in a simulation, it's impossible to prove them wrong.
To get back to synchronicity, if my worldview is already theistic, or idealistic, apparent cases of synchronicity just confirm what I already suspected (or had no reason not to suspect): sometimes people read each other's minds (or a bit of code in the Matrix is shared by two people, or a part of the dream we're all in, whatever). If my worldview is materialistic, synchronicity is nothing more than coincidence. In the end, you can't convince me materialism is the preferred -ism anymore than I can convince you that this is all a dream.
Which is why these threads where people are so sure of themselves are so amusing.