 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 $t$ isThe probability at least one has failed isThe 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 $\max (0,F_1+F_2-1)\le P(\text{both})\le \min (F_1,F_2)$, and inclusion-exclusion gives $P(\text{at least one})=F_1+F_2-P(\text{both})$.