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

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.

ii

For

p > 0

, equivalently

0 \leq q < 1

n = 1 \sum \infty q^{n - 1} p = \frac{p}{1 - q} = 1.

Thus absorption probability is one. The qualification matters: if

p = 0

, the process alternates between

B

and

I

forever and never reaches

A

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