## Problem 209 — rank:244

**Title:** Identifiability of stages in zoonotic spillover

**Added binding assumptions (LaTeX):**

```latex
Condition all stage probabilities on the same recorded-exposure population. Define $B$ as an underlying qualifying cluster outcome, observed through the detection mechanism, so it does not already imply $O=1$ by definition. The product additionally requires conditional independence of detection and cluster outcome given infection, exposure and $X$, with no false-positive branch unless explicitly included. Without those restrictions retain the general joint probability rather than the three-factor identity.
```

**Direct edits:** 3. **Fresh external checks:** None; supplied sources retained as provenance.