  Let $LA_{n}(\tau)$ be the length of the longest alternating subsequence of a
uniform random permutation $\tau\in[n]$. Classical probabilistic arguments are
used to rederive the asymptotic mean, variance and limiting law of
$LA_{n}(\tau)$. Our methodology is robust enough to tackle similar problems for
finite alphabet random words or even Markovian sequences in which case our
results are mainly original. A sketch of how some cases of pattern restricted
permutations can also be tackled with probabilistic methods is finally
presented.
