  We consider the best-choice problem for independent (not necessarily iid)
observations $X_1, \cdots, X_n$ with the aim of selecting the sample minimum.
We show that in this full generality the monotone case of optimal stopping
holds and the stopping domain may be defined by the sequence of monotone
thresholds. In the iid case we get the universal lower bounds for the success
probability. We cast the general problem with independent observations as a
variational first-passage problem for the running minimum process which
simplifies obtaining the formula for success probability. We illustrate this
approach by revisiting the full-information game (where $X_j$'s are iid
uniform-$[0,1]$), in particular deriving new representations for the success
probability and its limit by $n \rightarrow \infty$. Two explicitly solvable
models with discrete $X_j$'s are presented: in the first the distribution is
uniform on $\{j,\cdots,n\}$, and in the second the distribution is uniform on
$\{1,\cdots, n\}$. These examples are chosen to contrast two situations where
the ties vanish or persist in the large-$n$ Poisson limit.
