= Solution
Under the proposed exposure-only model, suppose a unit of infectious exposure has the same probability of producing an ascertained onset in either cohort. Conditional on the total of $108$ onsets, the allocation is a <binomial distribution> with younger-cohort probability
$$
\pi=\frac{85000}{85000+170000}=\frac13.
$$
Thus the expected cohort counts are $36$ and $72$, respectively, whereas the observed allocation is reversed. The <Pearson chi-squared goodness-of-fit test> gives
$$
X^2=\frac{(72-36)^2}{36}+\frac{(36-72)^2}{72}=54.
$$
With one degree of freedom its approximate <p-value> is $2.0\times10^{-13}$; the exact upper <binomial distribution> tail $\Pr\{\operatorname{Bin}(108,1/3)\geq72\}$ is approximately $1.56\times10^{-12}$. These are exceptionally incompatible with proportional allocation. Equivalently, the observed onset count per exposure unit is four times larger in the younger cohort:
$$
\frac{72/85000}{36/170000}=4.
$$
\b[Under the stated common-risk exposure model, the observed counts are not consistent with the early exposure period as their sole explanation.] This does not identify the source of the discrepancy, or prove a particular later exposure caused it. The inference also requires comparable ascertainment and a common fraction of exposure-derived cases manifesting by the observation date. Age-independent <incubation periods> alone do not guarantee that fraction when the two cohorts' exposure profiles within the decade differ; decade totals omit that timing information. The reasoning concerns the hypothetical historical <BSE>/<vCJD> model, not an estimate of current disease risk.
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