Lemma. Union bound for repeated false accepts [ftip-00E6]
Lemma. Union bound for repeated false accepts [ftip-00E6]
For finitely many candidate events from Definition [ftip-00E5],
\[\Pr \left (\bigcup _{i=1}^{N}F_i\right )\le \sum _{i=1}^{N}\Pr (F_i) =\sum _{i=1}^{N}q_i\eta _i.\]This is the union bound and requires no independence. If every marginal is at most \(\eta \), the right side is at most \(N\eta \).