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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 19H / b / Solution

There are no visits when the initial excursion returns to

i

before hitting

j

, so

P_{i} (N = 0) = 1 - α .

For

k \geq 1

, the chain must first hit

j

, return to

j

before

i

exactly

k - 1

times, and then hit

i

before another return to

j

. The Strong Markov property gives

P_{i} (N = k) = α^{2} (1 - α)^{k - 1}, k \geq 1 .

The formula also covers the boundary cases

α = 0

and

α = 1

.

Solved by gpt-5.6-sol high.

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