In general we will fix a prompt \(x\) and generator \(\pi\), which includes any decoding and stopping settings. Each output \(Y\) is assumed to include its reasoning and final answer, and the function \(g(Y)\) extracts that answer.
We draw \(N\) candidates independently from the generator: \[ Y_1,\ldots,Y_N\stackrel{\mathrm{iid}}{\sim}\pi(\cdot\mid x), \qquad N\geq1\text{ fixed before sampling}. \]
Write \(Y_{1:N}=(Y_1,\ldots,Y_N)\) for the overall candidate pool.
The selector returns one candidate unchanged: \[ Y^\star=\operatorname{Sel}_N(x;Y_{1:N})\in\{Y_1,\ldots,Y_N\}, \]
and we write the served-output distribution as \(Y^\star\sim q_N(\cdot\mid x)\) where \(q_N\) includes any selector randomness.
Correctness \(c(x,y)\in\{0,1\}\) is fixed independently of the model or verifier and depends on \(y\) only through \(g(y)\). A correct answer need not have valid reasoning. The selector need not know \(c\).
Accuracy and coverage. For any output law \(\mu\), define \[ \begin{aligned} a_\mu(x)&:=\mathbb E_{Y\sim\mu(\cdot\mid x)}[c(x,Y)],\\[0.5em] \operatorname{Cover}_N(x)&:=\Pr\!\left(\exists i\leq N:c(x,Y_i)=1\right)\\ &=1-[1-a_\pi(x)]^N. \end{aligned} \] Thus \(a_\pi\) is generator accuracy and \(a_{q_N}\) is served accuracy. All probabilities condition on \(x\).
Error decomposition. \[ 1-a_{q_N}(x)=\underbrace{1-\operatorname{Cover}_N(x)}_{\text{generation loss}}+\underbrace{\operatorname{Cover}_N(x)-a_{q_N}(x)}_{\text{selection loss}\;\geq\;0}. \]
Population averages. For prompts \(X\sim\mathcal D\) and a common budget \(N\), \[ \begin{aligned} \operatorname{pass@N}&:=\mathbb E_{X\sim\mathcal D}[\operatorname{Cover}_N(X)],\\ \text{served accuracy}&:=\mathbb E_{X\sim\mathcal D}[a_{q_N}(X)]. \end{aligned} \]