How to Solve Discrete Random Variables — Univariate RV — pmf defined by a geometric series Questions on Exam P

The constant successive-probability ratio identifies a geometric distribution on nonnegative integers. Normalization determines its initial mass, after which its tail and mean follow from a geometric series. Setting \(r=1/3\), the recurrence \(p_{n+1}=r p_n\) yields \(p_n=p_0 r^n\) for \(n\ge 0\). Normalizing probabilities gives \(\sum_{n=0}^\infty p_n=p_0\sum_{n=0}^\infty r^n=\dfrac{p_0}{1-r}=1\implies p_0=1-r=1-\dfrac{1}{3}\). Thus \(N\sim\operatorname{Geometric}(1-r)\) on \(\{0, 1, 2, \ldots\}\) with parameter \(r=1/3\). The tail probability is \(\Pr(N>1)=\sum_{n=2}^\infty (1-r)r^n=r^2=\left(\dfrac{1}{3}\right)^2=\frac{1}{9}\). Thus \(\Pr(N>1)=\frac{1}{9}\), which matches the correct option.