Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / b3

Table of contents
- Solution b3

Solution

A closed circuit surrounding the origin and passing through

(m, 0)

has diameter at least

m

. By the one-point estimate used in part (b2), its probability is at most

C e^{- c m}

. Therefore

P (\exists m \geq n on such a circuit) \leq \sum_{m \geq n} C e^{- c m} \leq C^{'} e^{- c^{'} n} .

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