  A border of a string is a non-empty prefix of the string that is also a
suffix of the string, and a string is unbordered if it has no border other than
itself. Loptev, Kucherov, and Starikovskaya [CPM 2015] conjectured the
following: If we pick a string of length $n$ from a fixed non-unary alphabet
uniformly at random, then the expected maximum length of its unbordered factors
is $n - O(1)$. We confirm this conjecture by proving that the expected value
is, in fact, ${n - \Theta(\sigma^{-1})}$, where $\sigma$ is the size of the
alphabet. This immediately implies that we can find such a maximal unbordered
factor in linear time on average. However, we go further and show that the
optimum average-case running time is in $\Omega (\sqrt{n}) \cap O (\sqrt{n
\log_\sigma n})$ due to analogous bounds by Czumaj and G\k{a}sieniec [CPM 2000]
for the problem of computing the shortest period of a uniformly random string.
