  Random walks, and in particular, their first passage times, are ubiquitous in
nature. Using direct enumeration of paths, we find the first return time
distribution of a 1D random walker, which is a heavy-tailed distribution with
infinite mean. Using the same method we find the last return time distribution,
which follows the arcsine law. Both results have a broad range of applications
in physics and other disciplines. The derivation presented here is readily
accessible to physics undergraduates, and provides an elementary introduction
into random walks and their intriguing properties.
