 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/28j/b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The Markov property and conditioning on the intermediate state give the bridge probability

P (X_{t} = 2 ∣ X_{0} = 1, X_{3 t} = 1) = \frac{p _{12} ( t ) p _{21} ( 2 t )}{p _{11} ( 3 t )} .

Substitution from part (i) yields

\frac{λ μ ( 1 - e ^{- s t} ) ( 1 - e ^{- 2 s t} )}{s ( λ + μ e ^{- 3 s t} )} (s = λ + μ) .

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