 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/ib/paper-1/20h/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-07

Let

T_{i}

be the expected time to complete the specified directed transition, starting at

i

c

. First-step decomposition gives

T_{a} = 1 + \frac{3}{5} T_{b} + \frac{2}{5} T_{c}, T_{b} = 1 + \frac{3}{4} T_{a}, T_{c} = 1 + \frac{2}{3} T_{a} + \frac{1}{3} T_{b} .

The missing

b \to c

continuation in the middle equation represents successful completion, but its step is counted by the leading one. Substitution gives

T_{a} = 128/11, T_{b} = 107/11, T_{c} = 12.

Thus the requested expectation from

a

128/11

steps.

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