Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 4 / a / i / Solution

Put $q=1-p$. Since $X$ takes positive integer values, $\mathbb{P}(X>0)=1=q^0$. Taking differences of the given tail probabilities therefore yields, also for $k=1$,This is the probability mass function of the geometric distribution on $\{1,2,\dots \}$.