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Joint coverage in detection-calibrated prevalence bounds controls the chance that both the observed-event bound and its detection calibration are valid. A prevalence calculation that substitutes an estimated detection probability into a sampling bound does not automatically retain the original confidence level.

In an original model with no false detections, prevalence $p$ and detection probability $q>0$ give observed-event probability $r=qp$. Let $r_U$ be an upper confidence bound with coverage at least $1-\alpha$, and $q_L>0$ a lower confidence bound with coverage at least $1-\beta$. On the event that both bounds cover,The union bound gives $\mathrm{Pr}(r>r_U\text{ or }q<q_L)\le \alpha +\beta$. Hence $p_U=\min \{1,r_U/q_L\}$ has coverage at least $1-\alpha -\beta$. This argument does not require independence between the two estimates, but it does require valid marginal bounds for the stated model. If no positive lower bound on $q$ is established, the resulting prevalence bound may be uninformative.

Exact extreme-count examples follow by inverting binomial distribution probabilities. NIST describes exact interval construction from binomial tails and distinguishes one-sided limits.^[ref] Zero detections among $n$ specimens gives $r_U=1-{\alpha }^{1/n}$. Detecting all $m$ independently known positive calibration specimens gives $q_L={\beta }^{1/m}$. These are one-sided extreme-case formulas derived from $(1-r{)}^n$ and $q^m$.

For stipulated $n=37$, $m=20$ and $\alpha =\beta =0.025$, the combined coverage is at least 95% and $p_U\approx 0.114$. Choosing two separate 95% bounds instead guarantees only at least 90% joint coverage by this argument; if their failures are independent, the joint success probability is $0.95^2=0.9025$. Neither is a 95% joint guarantee.

Error-budget allocation between prevalence and detection calibration need not split the allowed failure probability equally. A prespecified allocation can reflect which experiment supplies the weakest information. Calibration specimens must also represent the detection difficulty of the target specimens; a formally valid interval for an easier calibration set does not establish the relevant $q$.