Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / a / i / Solution

With independent Bernoulli trials of success probability $p$, $N$ counts the failures before the first success. There are $n$ failures followed by one success with probability $(1-p{)}^{n}p$, giving the stated geometric distribution on $\{0,1,2,\dots \}$. It counts failures, rather than the total number of trials including the success.