These sort of questions pop up regularly in the research of a colleague at work. He studies long polymer chains of length $L$ and carbon-carbon length $a$ that are in a liquid.
What is the probability that such a polymer will go round a tube of radius $R$? It is assumed that $a << R < L$.
The problem is not well posed, most probably. What else is necessary to answer? (One could look for a path that is almost closed, within a distance $h$.)