 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/4/b/solution

Write

q = 35/36

. Alice wins on her

(n + 1)

st turn if

n

complete rounds fail and then she succeeds. These disjoint events give the geometric series

P_{A} = \sum_{n = 0}^{\infty} q^{2 n} p = \frac{p}{1 - q ^{2}} = \frac{1}{1 + q} .

Consequently

P_{A} = \frac{36}{71} .

The probability of never obtaining a winner is

lim_{n \to \infty} q^{2 n} = 0

. The small advantage over

1/2

comes from taking the first turn.

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