 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/5/d/i/solution

Write

q = P_{B I}

and

p = P_{A I} = 1 - q

. A path with

n

visits to

I

makes

n - 1

unsuccessful excursions back to

B

, followed by one successful step to

A

. Every

B \to I

step has probability one, so

Pr (N = n) = q^{n - 1} p, n = 1, 2, \dots .

The visit count has a geometric distribution, with success meaning the final

I \to A

step.

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