Species richness

Suppose on the other hand that $R(\pi_k) = {1 - \pi_k \over \pi_k}$, the probability that the next species you encounter is different from the one you have just seen relative to the probability of encountering the same species again. Then

$$\begin{eqnarray*} \Delta_{-1}(C) &=& \sum_{k=1}^s\pi_k\left({1 - \pi_k \over \pi_k}\right) \cr &=& s - 1 \cr \end{eqnarray*}$$

We subtract one from the number of species since a community with only one species has zero diversity.