For such a problem, usually
&quot;at random&quot; means a uniform
distribution.  But on
the plane, there is no
uniform distribution.
So, the &quot;paper&quot; can&#x27;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.