  This paper considers a variation of the full-information secretary problem
where the random variables to be observed are independent but not necessary
identically distributed. The main result is a sharp lower bound for the optimal
win probability. Precisely, if $X_1,\dots,X_n$ are independent random variables
with known continuous distributions and $V_n(X_1,\dots,X_n):=\sup_\tau
P(X_\tau=M_n)$, where $M_n:=\max\{X_1,\dots,X_n\}$ and the supremum is over all
stopping times adapted to $X_1,\dots,X_n$, then $$V_n(X_1,\dots,X_n)\geq
\left(1-\frac{1}{n}\right)^{n-1},$$ and this bound is attained. The method of
proof consists in reducing the problem to that of a sequence of two-valued
random variables, and then applying Bruss' sum-the-odds theorem (2000). In
order to obtain a sharp bound for each $n$, we improve Bruss' lower bound
(2003) for the sum-the-odds problem.
