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Equation 4 · Part 6 · Ten Failure Modes That Define Deployed AI Safety Systems

superscript

P(at least one success in k tries)=1−(1−p)k.P(\text{at least one success in } k \text{ tries}) = 1 - (1-p)^k.
superscript

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

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(at least one success in k tries)=1−(1−p)kP(\text{at least one success in } k \text{ tries}) = 1 - (1-p)^k. At p =…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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