Let $A$ be an arbitrary list with $n$ distinct elements. Suppose I have a separate list, $B$, which is initially identical to $A$. Denote individual elements in $B$ as $b_k$ where $k\in\{1,2,...,n\}$. Now suppose I perform an experiment in which I randomly choose an element from $B$ (each element in $B$ is chosen with fixed, non-zero probability $P_{b_k}$ where $\sum_{k=1}^n P_{b_k}=1$), remove the corresponding element from $A$, and then continue until $A$ is empty. Since items were not removed from $B$, an already-removed item from $A$ could be chosen again. Let the random variable $X$ be the number of attempts it takes to empty the list $A$. What is the distribution of $X$ as dependent on $n$ and the $b_k$?
We know that the support of $X$ is $\{n,n+1, n+2, ...\}$, but it seems very confusing to assign these probabilities for each $X$, especially since each element in $B$ is not equally likely to be chosen. What is the probability mass function of $X$? I think that this is a similar problem to the "Coupon collector's problem," except here the probabilities aren't equally likely.
This is indeed a generalization of the coupon collector's problem. I strongly suspect that any attempt to write down the entire probability distribution for $X$ will be cumbersome and unenlightening. Here's a question whose accepted answer shows how to calculate the expectation of $X$: Non-standard coupon collector's problem?
The Wikipedia article on the coupon collector's problem provides this expression for the expectation of $X$:
$$ E(X)=\int_0^\infty \big(1-\prod_{k=1}^n(1-e^{-P_{b_k}t})\big)\mathrm dt\;. $$
It's quoted from Flajolet, Philippe; Gardy, Danièle; Thimonier, Loÿs (1992), “Birthday paradox, coupon collectors, caching algorithms and self-organizing search”, Discrete Applied Mathematics, 39 (3): 207–229, doi:10.1016/0166-218X(92)90177-C, MR 1189469.
Here's another article that derives the same result and also has all sorts of other interesting stuff about the coupon collector's problem.
As far as I can tell, neither of these articles tries to set up the entire probability distribution for $X$.