What does this equation mean?

The failure modes built on repeated querying — jailbreak search, paraphrase evasion, mismatched-language generalisation — share a further structural property worth stating precisely, because a single strong per-attempt success rate is often reported as though it settled the question of robustness on its own. Suppose a safeguard blocks a fixed, independently generated attack attempt with probability 1-p , so a single try succeeds only with probability p . An adversary who is not limited to one try, and who runs k independent attempts — exactly the loop PAIR and the suffix-search method both implement mechanically [ 2 , 1 ] — succeeds at least once with probability $P(\text{at least one success in } k \text{ tries}) = 1 - (1-p)^k$. At p = 0.05 , a safeguard that looks 95% reliable per attempt, twenty tries already succeed with probability 1-(0.95)^{20} $\approx$ 0.64 , and by sixty tries cumulative success exceeds 0.95 — a per-attempt figure that reads as strong evidence of robustness collapses once retrying is free, which is precisely the black-box search loop several of the documented attacks above run automatically. This is not a result reported by any single cited study; it is the elementary shape of what those studies demonstrate empirically, and it is the reason a reported single-attempt block rate, on its own, answers a narrower question than it is usually taken to answer.