 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-2/10e/b/ii/solution

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

Part (i) gives

P (A) = 2/3

. Since

{X_{1} = 0} \subseteq A

1/2

P (X_{1} = 0 ∣ A) = \frac{1/2}{2/3} = \frac{3}{4} .

Conditioned on

A

, a walk at one moves to zero with probability

3/4

and to two with probability

1/4

. A walk at two must next move to one, since a move to three would violate

A

. If

e_{i} = E_{i} (T ∣ A),

e_{1} = 1 + \frac{1}{4} e_{2}, e_{2} = 1 + e_{1} .

Solving,

E (T ∣ A) = e_{1} = \frac{5}{3} .

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