For such a problem, usually "at random" means a uniform distribution. But on the plane, there is no uniform distribution. So, the "paper" can't be the plane. So, it might be fair to ask the size of the paper and what happens with the needle near the edges? E.g., on a rectangular sheet of paper of finite size, the needle can land in a position so that it does not cross a line but would on a larger sheet of paper.