I would like to know that probability of an item occurring in a multiset (a combination of selections with repetitions).
Given a set $S = \{x_1,x_2,...,x_n\}$ the number of possible unordered subsets of size $k$ that can be chosen with repetitions is:
$${\left(n\choose k\right)} = {n + k - 1\choose k}$$
So if I want to know the probability that any given $x$ occurs in at least one of these multisets, how do I find it?
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I'm trying to come up with a random variable $X_i$ which represents the probability that $x_i$ occurs at least once in a multiset.
If you pick one element uniformly at random from $S$, there is a probability of $\frac{n-1}n$ of not getting a fixed element $x \in S$. You do this $k$ times, the probability of not getting any $x$'s is $\left(\frac{n-1}{n}\right)^k$. Subtract this from $1$ to get the probability of getting at least one $x$.
I assume that the distribution of picking an element is uniform, but the distribution of multisets is not.
If you assume the distribution of multisets is uniform, then you would divide the number of sets having $x$ by the number of sets not having $x$. The number of multisubsets of $S$ of size $k$ not having $x$ is the same as the number of multisubsets of $S - \{x\}$ of size $k$, which is ${n + k - 2 \choose k}$. Therefore, the probability of not having $x$ in the multiset is $\frac{n + k - 2 \choose k}{n + k - 2 \choose k} = \frac{n-1}{n+k-1}$, and the answer to the question (the probability of finding $x$ at least once in the multiset) is $1-\frac{n-1}{n+k-1}=\frac k{n+k-1}$.