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Equation 4 · Part 2 · Ten Failure Modes That Define Deployed AI Safety Systems

Symbol k

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.
kk

What this part means

k is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

k is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

The passage around this formula

…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 = 0.05 , a safeguard that looks 95% reliable per attempt, twenty tries already succeed with probability 1-(0.95)^{20}…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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