= Failure probabilities for two independent lifetimes
For independent <survival times> $T_1,T_2$ with <cumulative distribution functions> $F_1,F_2$, the probability both components have failed by time $t$ is
$$
P(\max(T_1,T_2)\leq t)=F_1(t)F_2(t).
$$
The probability at least one has failed is
$$
P(\min(T_1,T_2)\leq t)=1-(1-F_1(t))(1-F_2(t)).
$$
The first equality uses <independence of random variables>; the second applies it to the complementary event that both survive. These are the distributions of the two extreme <order statistics>.
Marginal distributions alone do not determine these probabilities. Without independence, the <Fréchet inequalities> give $\max(0,F_1+F_2-1)\leq P(\text{both})\leq\min(F_1,F_2)$, and <inclusion-exclusion> gives $P(\text{at least one})=F_1+F_2-P(\text{both})$.
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