Rao-Blackwell estimator of a geometric failure probability ($\hat{p}=(T-n)/(T-1),T={\sum }_{j=1}^n{X}_j,n\ge 2$)

For independent geometric observations with mass $p^{r-1}(1-p)$, the total $T$ is sufficient. Conditional on $T=t$, positive-integer compositions are uniformly distributed. The indicator $1_{\{X_1>1\}}$ is unbiased for $p$, and conditioning it on $T$ gives $\left(\genfrac{}{}{0ex}{}{t-2}{n-1}\right)/\left(\genfrac{}{}{0ex}{}{t-1}{n-1}\right)=(t-n)/(t-1)$. For one observation, use $1_{\{T>1\}}$ instead, including its zero value at $T=1$.