Theorem. Independent false accepts amplify with candidate count [ftip-007L]
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    In the repeated audit of Definition [ftip-007K], suppose the pairs \((S_i,A_i)\) are independent and identically distributed. Assume

    \[ \iota =\Pr (S_i=0)>0, \qquad \eta _+=\Pr (A_i=1\mid S_i=0). \]

    Then the probability that at least one accepted invalid candidate exists among the \(N\) audited candidates is

    \[ \Pr (\mathcal E_N^{\rm fa})=1-(1-\iota \eta _+)^N. \]

    It is strictly increasing in \(N\) when \(0<\iota \eta _+<1\), and it converges to \(1\) when \(\iota \eta _+>0\).

    Proof. 

      Each \(C_i^{\rm fa}\) has probability \(\iota \eta _+\). Independence of the audited pairs makes the events \(C_i^{\rm fa}\) independent. Therefore

      \[ \Pr \left ((\mathcal E_N^{\rm fa})^c\right ) =\Pr \left (\bigcap _{i=1}^N (C_i^{\rm fa})^c\right ) =\prod _{i=1}^N(1-\iota \eta _+) =(1-\iota \eta _+)^N. \]

      Taking complements gives the equality. The monotonicity and limit follow from the elementary powers of \(1-\iota \eta _+\).

      This finite statement follows from the displayed hypotheses.