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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 $P(\text{at least one})=F_1+F_2-P(\text{both})$.

On the probability space $(X^2,\eta )$, put $A=I\times X$ and $B=X\times J$. The marginal conditions give $\eta (A)=x$ and $\eta (B)=y$, while $A\cap B=I\times J$. The elementary Fréchet inequalities yieldHence

The product formula requires independent failure times. This assumption is not stated in the original PDF. Under that intended assumption, put $s=t/T$ and use the failure probabilities for two independent lifetimes:Its graph is a cubic increasing from $(0,0)$ to $(T,1)$, convex before reaching its plateau.

Without independence, the marginal distributions do not determine $P(t)$. The Fréchet inequalities give the sharp pointwise boundsFor example, taking $t_1=TU$ and $t_2=T\sqrt{U}$ with the same uniform random variable $U$ preserves both prescribed marginals but gives $P(t)=s^2$, rather than $s^3$.