Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 7 / b / Solution

For $0<p<1$, independence of the remaining indicators givesBecause $0<k<n$, the latter probability is positive for interior $p$. The two events are independent events exactly when their joint probability equals $p\mathbb{P}(S_n=k)$, which is equivalent to $p=k/n$.

At $p=0$ or $p=1$, the event $\{S_n=k\}$ is null, and a null event is independent of every event. Therefore the full answer, including the degenerate endpoints, is