Consider the integers $\{1,\dots, N\}$ for some large $N$. Let us suppose that for each $\{1, \dots, n\}$ there are associated probabilities $p_1, \dots, p_n$ with $n \ll N$.
We sample independently and repeatedly from $\{1,\dots, N\}$. For each $i \leq n$ we sample the integer $i$ with probability $p_i$. With probability $q= 1-\sum_{1}^np_i$ we sample some integer greater than $n$. We are not concerned with which integer in this case.
After $s$ samples, how can you compute the expected number of distinct integers less than or equal to $n$ that have been sampled?
If anything is unclear, please let me know.
Let $X_i$ be the variable with two values: 1 if the number $i$ is sampled an $0$ if it is not sampled. Then the expected value of the variable after $s$ samples is just probability that the number $i$ is sampled: $$ \mathbb E(X_i)=1-(1-p_i)^s. $$
By linearity of expectation, the expected number of sampled numbers in the range $1\dots n$ is: $$ \mathbb E\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n\mathbb E(X_i)=n-\sum_{i=1}^n(1-p_i)^s. $$