The problem is ill-posed: What is the probability of choosing does not describe the choosing process. For example: It turns out that I always choose my 2 points so that the center-of-the-square is never inside the circle, so the probability is exactly 0.
Not the answer that anyone is looking for, but shows the importance of understanding what is asked.
Edit: and learning to specify exactly what you mean when you ask a question. As the saying goes: If you do not learn to write what you mean, then you will have trouble meaning what you write.
I completely agree with this and always find it instructive to examine wording thoroughly when posing problems or solutions. However, I think you could come across more likably by offering a better wording yourself in your comment. I'll have a shot at it:
"Given two random points inside a square, what is the probability that the center of the square lies in the circle formed by taking the points as diameter?"
I'm not happy with my opening clause there though, possibly because I omitted to explicitly state the distribution (that is, uniform - all points are equally likely). It seems there is a convention to assume uniform when no other information is given, but I still don't like "random" there. What would be your wording?
Not the answer that anyone is looking for, but shows the importance of understanding what is asked. Edit: and learning to specify exactly what you mean when you ask a question. As the saying goes: If you do not learn to write what you mean, then you will have trouble meaning what you write.