Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 8 / a / i / Solution

In independent Bernoulli trials with success probability $p$, $N$ counts failures before the first success. The event $N=n$ requires $n$ failures followed by one success, giving $(1-p{)}^{n}p$. This is the zero-based geometric distribution, rather than the count of trials including the success.