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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