= Finite-jump boundary crossing identity
{title2=$\mathbb P(1-S+X_t>t\text{ for all }t<1)=1-S$}
Let $U_1,\ldots,U_n$ be <independent> <uniform random variables> on $[0,1]$, let $\theta_i>0$, and set $S=\sum_i\theta_i\in(0,1)$ and $X_t=\sum_i\theta_i\mathbf1_{\{U_i\leq t\}}$. Then
$$
\mathbb P(1-S+X_t>t\text{ for all }0\leq t<1)=1-S.
$$
Indeed, the complementary crossing is that of level one by the <uniform-arrival residual mass martingale> $M_t=(S-X_t)/(1-t)$, with $M_0=S$. Its jumps are downward, so its first crossing of one is continuous. An arrival coincides with a possible crossing time with probability zero: the possible times belong to the finite deterministic set obtained from subsets of the weights. Stop at that crossing or at $r<1$. The stopped values are in $[0,1]$, and as $r\uparrow1$ they converge to the crossing indicator because all arrivals occur before one <almost surely>. The <optional stopping theorem> and <dominated convergence theorem> give crossing probability $S$. For $S=1$, the strict displayed inequality already fails at time zero.
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