I think your way of looking at it requires an explosion of alien civilisations to find one in which our random 1000 digits is their special number. If you generate 1 extra digit, you'd need 10x as many civilizations to expect a match.
Getting back to the experiment, obviously it's measuring the human mind so the context is the person's mind. They might enter their bank account number which looks random to the researchers but is actually cheating because they're not using their brain as the source of randomness.
That's an interesting point about the random sequence selected what's the chance it matches an alien civilization constant. I didn't think about that way. I guess I thought about it more from your other example point of view, where say an alien civilization pretends to be a human and gives you a number that they say is random (just like the guy giving you his bank account number) but actually the alien civilization gave you a physical constant.
It's interesting that there's a difference but it's sort of like the difference between you picking a random sequence from a random source and the probability of that being actually random versus you being given a supposedly random sequence by somebody and the probability of that being actually random. I think.
Yea I guess we couldn't tell if somebody gave us a number that was special to them but appeared random. Afterall, the alien's physical constant (or rather, the arbitrary definitions of their units) would have effectively been chosen by a random number generator just like ours are.
If you pick a random sequence from a random source, then it's truly random by definition, isn't it? Even if it happens to be 1111111. It might score badly on a randomness measure but no randomness measure is truly perfect.
Getting back to the experiment, obviously it's measuring the human mind so the context is the person's mind. They might enter their bank account number which looks random to the researchers but is actually cheating because they're not using their brain as the source of randomness.