  The sequential sampling of populations with unequal probabilities and with
replacement in a closed population is a recurrent problem in ecology and
evolution. Many of these questions can be reformulated as urn problems, often
as special cases of the coupon collector problem, most simply expressed as the
number of coupons that must be collected to have a complete set. We aimed to
apply the coupon collector model in a comprehensive manner to one example
-hosts (balls) being searched (draws) and parasitized (ball color change) by
parasitic wasps- to evaluate the influence of differences in sampling
probabilities between items on collection speed.
  Based on the model of a complete multinomial process over time, we define the
distribution, distribution function, expectation and variance of the number of
hosts parasitized after a given time, as well as the inverse problem,
estimating the sampling effort. We develop the relationship between the risk
distribution on the set of hosts and the speed of parasitization and propose a
more elegant proof of the weak stochastic dominance among speed of
parasitization, using the concept of Schur convexity and the "Robin Hood
transfer" numerical operation.
  Numerical examples are provided and a conjecture about strong dominance is
proposed. The speed at which new items are discovered is a function of the
entire shape of the sampling probability distribution. The sole comparison of
values of variances is not sufficient to compare speeds associated to two
different distributions, as generally assumed in ecological studies.
