The probability that two individuals selected at random (without
replacement) from a sample will belong to different species is given by
(Hurlbert, 1971, p.579, Eq. 3):
$$\Delta_1 = \sum_{i=1}^S (\frac{N_i}{N})(\frac{N-N_i}{N-1}) =
(\frac{N}{N-1})\Delta_2$$
where \(\Delta_2\) (Hurlbert, 1971, p.579, Eq. 4) is the
probability that two individuals selected at random (with
replacement) from a sample will belong to different species:
$$\Delta_2 = 1 - \sum_{i=1}^S \pi_i^2$$
where \(N_i\) is the number of individuals of the ith
species in the community, \(N\) is the total number of individuals in the
community, \(\pi_i = N_i/N\), and \(S\) is the number of
species in the community.
Note that Hurlbert's PIE hpie is the complement of
simpson.