Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 10E / b / i

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- Solution i

Solution

Let

h (x)

be the probability that the simple symmetric random walk starting at

x \in {0, \dots, n}

hits zero before

n

. The first-step recurrence is

h (x) = \frac{1}{2} h (x - 1) + \frac{1}{2} h (x + 1), h (0) = 1, h (n) = 0.

The recurrence says that

h

is affine, and the boundary values determine it:

h (x) = \frac{n - x}{n} .

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