Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 12 / b / ii

A forced plain choice occurs exactly when the last chocolate is taken before the final Fellow is served: if it is taken by choice $f\le F-1$, 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, henceEquivalently, 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 $P_{\mathrm{Disaster}}={\sum }_{k=C}^{F-1}\left(\genfrac{}{}{0ex}{}{F-1}{k}\right)p^k(1-p{)}^{F-1-k}$. The success-count and waiting-time duality explains equality of these two sums. Plain-biscuit depletion cannot interfere because $D>F$.