= Contribution-weighted precision target for inverse moments
{title2=$n\ge\theta^2(1-p)/(\eta^2p)$}
A \b[<contribution-weighted precision target for inverse moments>] chooses sample size according to how strongly rare observations affect the desired inverse moment, rather than according to the probability of seeing any rare observation. For positive thickness $H$, an inverse moment such as $\mu=\mathbb E(1/H)$ can be sensitive to very small thicknesses.
In an original two-point model, let reciprocal thickness be $a$ with rare probability $p$, and $b$ otherwise, with $a>b>0$. Then $\mu=b+(a-b)p$. For $n$ independent observations and rare count $N$, the estimate is $\widehat\mu=b+(a-b)N/n$. Since $N$ has a <binomial distribution>,
$$
\operatorname{Var}(\widehat\mu)=\frac{(a-b)^2p(1-p)}n.
$$
Define the fraction of the mean supplied by the rare excess over the baseline as $\theta=(a-b)p/\mu$. Its relative <standard uncertainty> is
$$
\frac{u(\widehat\mu)}{\mu}=\theta\sqrt{\frac{1-p}{np}},\qquad
n\ge\frac{\theta^2(1-p)}{\eta^2p}
$$
for target relative uncertainty $\eta$. Here $0<\theta<1$. Rarity alone does not specify the required precision because the rare value's magnitude also enters.
For stipulated $p=10^{-4}$ and $\eta=0.10$, a rare-excess contribution $\theta=0.05$ requires at least 2,500 observations, whereas $\theta=0.50$ requires 249,975. A tenfold increase in contribution raises this variance-based count a hundredfold. The first case still has approximately $(1-10^{-4})^{2500}=0.779$ probability of no rare observation. That is compatible with its standard-uncertainty target because missing the rare excess produces only a 5% relative error in that particular model.
This is not a 95% probability guarantee of a specified relative error. Discovery probability $1-(1-p)^n$, estimator <variance> and interval coverage are different criteria. Exact <binomial distribution> tail calculations address count probabilities; they do not turn a standard deviation into a coverage statement automatically.<NIST and SEMATECH e-Handbook of Statistical Methods>[§7.2.4.1, exact binomial-tail calculations]{ref}
The calculation also treats $a$, $b$ and $p$ as specified planning quantities. Uncertainty in rare-event probability, imaging resolution or the reciprocal values requires further modeling; spatial dependence reduces the usefulness of an independent-sample count. In particular, increasing $n$ cannot establish an assumed physical lower-thickness bound that the measurement system never resolves.
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