 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/9/b/iv/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 9 / b / iv / Solution by

Codex  0 2026-10-25

For

1 \leq i \leq k

, the success probability is

i / k > 0

, and the definition of conditional expectation on an event gives

E_{i} [X ∣ J = 1] = \frac{E _{i} [ X 1 _{{J = 1}} ]}{P _{i} ( J = 1 )} = \frac{i ( k ^{2} - i ^{2} ) / ( 3 k )}{i / k} .

(1)

Therefore

E_{i} [X ∣ reach k] = \frac{k ^{2} - i ^{2}}{3} minutes .

(2)

i = k

the value is zero because the trial has already ended. At the left absorbing state

i = 0

, conditioning on success would be undefined because it has zero probability. The formula concerns the successful trajectories, so it need not equal the unconditional mean

i (k - i)

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