  Consider a random permutation $\pi\in{\cal S}_n$. In this paper, perhaps best
classified as a contribution to discrete probability distribution theory, we
study the {\it first} occurrence $X=X_n$ of a I-II-III-pattern, where "first"
is interpreted in the lexicographic order induced by the 3-subsets of
$[n]=\{1,2,...,n\}$. Of course if the permutation is I-II-III-avoiding then the
first I-II-III-pattern never occurs, and thus $\e(X)=\infty$ for each $n$; to
avoid this case, we also study the first occurrence of a I-II-III-pattern given
a bijection $f:{\bf Z}^+\to{\bf Z}^+$.
