  What is the probability that a needle dropped at random on a set of points
scattered on a line segment does not fall on any of them? We compute the exact
scaling expression of this hole probability when the spacings between the
points are independent identically distributed random variables with a
power-law distribution of index less than unity, implying that the average
spacing diverges. The theoretical framework for such a setting is renewal
theory, to which the present study brings a new contribution. The question
posed here is also related to the study of some correlation functions of simple
models of statistical physics.
