= Rare-event discovery versus inverse-moment precision
\b[Rare-event discovery versus inverse-moment precision] separates seeing an influential thin region from estimating its contribution to inferred <density>. In a two-point thickness model, let a rare thickness occur with <probability> $p$, with reciprocal thickness $a$, while common regions have reciprocal thickness $b$. For $n$ independent observations, the rare count has a <binomial distribution>. The chance of at least one rare observation is
$$
P_{\rm discover}=1-(1-p)^n.
$$
To make this at least $1-\alpha$, choose $n\ge\log(\alpha)/\log(1-p)$. For stipulated $p=10^{-4}$ and $\alpha=0.05$, the smallest integer is 29,956. This planning statement assumes the specified rare-event <probability>; it does not estimate that probability from a sample that missed the event.<NIST and SEMATECH e-Handbook of Statistical Methods>[§7.2.4.1, exact binomial-tail construction]{ref}
The <sampling distribution> of the estimated inverse moment answers another question. With $N$ the observed rare count,
$$
\widehat\mu=b+(a-b)N/n,\qquad
\mu=b+(a-b)p,\qquad
\operatorname{Var}(\widehat\mu)=\frac{(a-b)^2p(1-p)}n.
$$
Thus a target relative <standard uncertainty> $\eta$ for the inverse moment requires
$$
n\ge\frac{(a-b)^2p(1-p)}{\eta^2\mu^2}.
$$
This is a <contribution-weighted precision target for inverse moments>: the required count depends on both rarity and the leverage of the rare value.
Use the original model $a=1000\,\mathrm{\mu m^{-1}}$, $b=1/10.001\,\mathrm{\mu m^{-1}}$ and $p=10^{-4}$. Here $\mu\approx0.19998\,\mathrm{\mu m^{-1}}$ and $(a-b)p/\mu=1/2$. Consequently
$$
\frac{u(\widehat\mu)}{\mu}=\frac12\sqrt{\frac{1-p}{np}}.
$$
At the 95%-discovery count, this relative <standard uncertainty> is still about 28.9%. Reaching 10% for the inverse moment needs 249,975 observations in this model, while reaching 10% for the rare fraction itself needs 999,900. The factor of four arises because the rare excess accounts for half the total inverse moment, so its relative fluctuation is diluted in the total.
These are exact <variance> calculations under a stipulated distribution, not confidence-coverage guarantees or measured material properties. Spatially clustered thin regions reduce the effective independent sample count. A credible physical lower-thickness bound or better resolved imaging can change the model more substantially than merely extending a simulation with an unsupported tail.
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