I have a model where I need to sample from $m$ elements without replacement, where each element exists exactly once. The element have different weights and the probability of an element being drawn is equal to the fraction of the element's weight relative to the sum of the weights of the remaining elements. In R a single draw would simply be:
m <- 4 # number fo elements w <- c(.4,.3,.2,.1) # weights n <- 3 # number of draws sample(seq_along(w), size = n, prob=w, replace=FALSE) >  3 1 2
Simulating many draws and calculating the probability of occurrence of each element we get
v <- replicate(1e5, sample(seq_along(w), size = n, prob=w, replace=FALSE)) p.hat <- table(v) / sum(table(v)) p.hat > 1 2 3 4 0.3075767 0.2902900 0.2526967 0.1494367
What is the name of the resulting distribution and how could I calculate the results above without simulation?
PS. I stumbled across Wallenius' noncentral hypergeometric distribution but my knowledge in this field is too limited as to decide if this distribution is given or not.