Actually, you would have a hard time producing this set in such way that no "circles" like that are found at all. It would have to be a very artificial distribution of points in space for you not to observe this, like all of them arranged in a single line, or a giant rectangle, idk.
It depends on the size of the circle, though. The smaller the size, the more likely the probability is. But that’s only for a particular combination of 50 dots. Now we have to average out of all possible circle sizes and all combinations of 50 dots. Can someone do the math (or the simulation)?
I'm thinking, the larger the space, the larger the number of points contained within it, so the larger the probability of them being arrange in such way that blah blah ...
We need a math guy to chime in. I have a hunch there may be a theorem about something like this already.