  We investigate the behavior of the periods and border lengths of random words
over a fixed alphabet. We show that the asymptotic probability that a random
word has a given maximal border length $k$ is a constant, depending only on $k$
and the alphabet size $\ell$. We give a recurrence that allows us to determine
these constants with any required precision. This also allows us to evaluate
the expected period of a random word. For the binary case, the expected period
is asymptotically about $n-1.641$. We also give explicit formulas for the
probability that a random word is unbordered or has maximum border length one.
