 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/f/ii/solution

The same geometric series identity gives

T_{A B} = \frac{p ( τ _{B} + τ _{I} )}{( 1 - q ) ^{2}} = \frac{τ _{B} + τ _{I}}{p} .

A direct renewal check gives the identical result:

T_{B} = τ_{B} + T_{I}

and

T_{I} = τ_{I} + q T_{B}

, so

p T_{B} = τ_{B} + τ_{I}

. This also checks the units: mean first-passage time has units of time, unlike the dimensionless step count.

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