Summary: For a given positive number ‘\(\delta\)’, we consider a sequence of \(\delta\)-neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a \(U(0,1)\) distribution, and “stop as soon as their union contains the interval \((0,1).\)” We call such a union “a cover.” To find the distributions of \(N(\delta)\), the stopping time random variable, we need the joint distribution of order statistics from a \(U(0,1)\) distribution. For each \(\delta>0\) and \(n=1,2,\ldots\), we obtain a general expression for \(P(N(\delta)\leq n)\), and for a fixed value of \(\delta\), it is the distribution function of \(N(\delta).\) For a given \(n\), let \(\Delta(n)\) be the minimum value of \(\delta\), so that the union of the \(n\) \(\delta\)-neighborhoods of the first \(n\) observations contains the interval \((0,1).\) Because \(N(\delta)\leq n\) if and only if \(\Delta(n)\leq \delta\), the distributions of \(\Delta(n)\) can be obtained by fixing \(n\) in the general expression for \(P(N(\delta)\leq n).\) To describe the impact of \(\delta\) on the distribution of \(N(\delta)\) and that of \(n\) on \(\Delta(n)\), we sketch the graphs of distribution functions and the empirical distribution functions.