Corollary. A rollout budget for target discovery probability [ftip-0079]
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    This is a finite corollary of Theorem [ftip-0078], obtained from its displayed hypotheses.

    Let \(0<\delta <1\). Under the independent and identically distributed setup of Theorem [ftip-0078], suppose first that \(0<p<1\). The least positive integer budget whose discovery failure probability is at most \(\delta \) is

    \[ B_{\min } =\left \lceil \frac {\log \delta }{\log (1-p)}\right \rceil . \]

    If \(p=0\), no finite positive budget reaches failure probability below one. If \(p=1\), one attempt suffices.

    Proof. 

      For \(0<p<1\), Theorem [ftip-0078] gives failure probability \((1-p)^B\). The inequality \((1-p)^B\leq \delta \) is equivalent to \(B\geq \frac {\log \delta }{\log (1-p)}\) because both logarithms are negative. The least integer solution is the displayed ceiling. The two boundary cases follow directly from \((1-p)^B\).