= Solution
A forced plain choice occurs exactly when the last chocolate is taken before the final Fellow is served: if it is taken by choice $f\leq F-1$, choice $f+1$ has no chocolate available, whatever that person's preference would have been. Exhaustion precisely on choice $F$ forces nobody. The distinct possible exhaustion times are disjoint events, hence
$$
\boxed{P_{\rm Disaster}=\sum_{f=C}^{F-1}\binom{f-1}{C-1}p^C(1-p)^{f-C}.}
$$
Equivalently, imagine independent preferred choices even after stock is exhausted. The event is that at least $C$ preferred chocolate choices occur among the first $F-1$ people, so the <binomial distribution> gives $P_{\rm Disaster}=\sum_{k=C}^{F-1}\binom{F-1}{k}p^k(1-p)^{F-1-k}$. The <success-count and waiting-time duality> explains equality of these two sums. Plain-biscuit depletion cannot interfere because $D>F$.
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