= Solution
Surveillance should be calibrated to the prevalence of concern, $q_*=1/200000=5\times10^{-6}$, rather than only to the higher rate used in the preceding calculation. With $50000$ representative independent tests,
$$
\Pr(\text{at least one positive}\mid q_*)=1-(1-q_*)^{50000}\simeq0.2212.
$$
Thus \b[50,000 tests give only about 22% detection probability at the concerning rate]; a negative survey is quite likely, with probability about $78\%$. It costs $50000\times\pounds40=\pounds2$ million. Even $500000$ tests, costing $\pounds20$ million, give about $91.8\%$ detection probability at $q_*$.
For a target $95\%$ detection probability, solve
$$
(1-q_*)^n\leq0.05,
\qquad
\boxed{n\geq\left\lceil\frac{\log0.05}{\log(1-q_*)}\right\rceil=599145.}
$$
The cost is about $\pounds24$ million. The <zero-event binomial upper confidence bound> gives the same perspective: after zero positives in $50000$ tests, the one-sided $95\%$ upper bound is
$$
1-0.05^{1/50000}\simeq5.99\times10^{-5},
$$
roughly one in $16700$, well above the concerning rate. \b[A 50,000-animal survey is insufficient to rule out that prevalence with high confidence.] A practical design should specify its desired detection probability, diagnostic sensitivity, representativeness and dependence between sampled animals. Targeting the higher-risk <scrapie>-positive stratum may improve efficiency if testing that stratum is feasible and its contribution to the slaughter population is estimated; its result then needs the appropriate population weighting. A cost alone does not establish adequate surveillance.
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