 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/14/b/ii/solution

The event includes stopping positions zero through

k

. Applying Bayes' theorem to their joint and marginal probabilities gives

P (G ∣ X \leq x_{k}) = \frac{p \sum _{j = 0}^{k} g _{j}}{p \sum _{j = 0}^{k} g _{j} + ( 1 - p ) \sum _{j = 0}^{k} d _{j}} .

This conditional probability is defined whenever its conditioning event has positive probability. At

k = n

, it reduces to

p

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