 Solution

ID: 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

n - 1

choices must include exactly

C - 1

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 gives

P_{LastChoc, n} = (C - 1 n - 1) p^{C} (1 - p)^{n - C}, C \leq n \leq F .

This 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 \leq n \leq F

. At

p = 0

the last chocolate is never taken, and at

p = 1

it is taken on choice

C

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