  We study the problem of maximizing the probability that (i) an electric
component or financial institution $X$ does not default before another
component or institution $Y$ and (ii) that $X$ and $Y$ default jointly within
the class of all random variables $X,Y$ with given univariate continuous
distribution functions $F$ and $G$, respectively, and show that the
maximization problems correspond to finding copulas maximizing the mass of the
endograph $\Gamma^\leq(T)$ and the graph $\Gamma(T)$ of $T=G \circ F^-$,
respectively. After providing simple, copula-based proofs for the existence of
copulas attaining the two maxima $\overline{m}_T$ and $\overline{w}_T$ we
generalize the obtained results to the case of general (not necessarily
monotonic) transformations $T:[0,1] \rightarrow [0,1]$ and derive simple and
easily calculable formulas for $\overline{m}_T$ and $\overline{w}_T$ involving
the distribution function $F_T$ of $T$ (interpreted as random variable on
$[0,1]$). The latter are then used to charac\-terize all non-decreasing
transformations $T:[0,1] \rightarrow [0,1]$ for which $\overline{m}_T$ and
$\overline{w}_T$ coincide. A strongly consistent estimator for the maximum
probability that $X$ does not default before $Y$ is derived and proven to be
asymptotically normal under very mild regularity conditions. Several examples
and graphics illustrate the main results and falsify some seemingly natural
conjectures.
