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 / i / Solution

For the last chocolate to be taken on choice $n$, the first $n-1$ choices must include exactly $C-1$ chocolate choices, and choice $n$ must take chocolate. Before that final success there is still chocolate available, so no forced plain choice has yet occurred. The negative binomial stopping argument givesThis is the all-trials convention of the negative binomial distribution. The finite examination scenario also has a remaining probability that the tray still contains chocolate after all $F$ choices; the displayed probabilities need not sum to one over $C\le n\le F$. At $p=0$ the last chocolate is never taken, and at $p=1$ it is taken on choice $C$.