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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 10F / d / Solution

When

r

red marbles remain, the next draw reduces their number with probability

r / n

. The waiting time

W_{r}

for that reduction has a geometric distribution on

{1, 2, \dots}

and therefore

G_{W_{r}} (s) = \frac{( r / n ) s}{1 - ( 1 - r / n ) s} = \frac{rs}{n - ( n - r ) s} .

The successive waiting times are independent, and

T = W_{n} + W_{n - 1} + \dots + W_{1} .

Using the product rule for probability-generating functions,

G_{T} (s) = r = 1 \prod n \frac{rs}{n - ( n - r ) s} .

Solved by gpt-5.6-sol high.

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