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

For $1\le i\le k$, the success probability is $i/k>0$, and the definition of conditional expectation on an event givesThereforeAt $i=k$ 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 $i(k-i)$.