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For each rational prime $p$ let $\mathbf{X}_p$ denote the random variable uniformly distributed in $\{0, 1, ..., p-1\}$, with all the $\mathbf{X}_p$ independent of each other. Define the coset $\mathbf{C}_p$ of the prime ideal $p\mathbb{Z}$ via

$$ \mathbf{C}_p=p\mathbb{Z}+\mathbf{X}_p.$$

Let $\mathbf{S}$ be the random variable equal to the size of the complement of the union of all the $\mathbf{C}_p$:

$$\mathbf{S} = \mathrm{card}(\mathbb{Z} - (\mathbf{C}_2 \cup \mathbf{C}_3 \cup \mathbf{C}_5 \cup \cdots)).$$

What is the expected value of $\mathbf{S}$?

(In the simplest case with all $\mathbf{X}_p=0$, we get $\mathbf{S}=2$.)

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    $\begingroup$ Apologies if this is making a basic mistake. But it seems to me that, for any fixed $n \in \mathbb{Z}$, the probability that $n$ is not in any of the $\mathbf{C}_p$ is $\prod_p (p-1)/p=0$. So by linearity of expectation, we have that the expected value of $\mathbf{S}$ is $0$. $\endgroup$
    – Sam Hopkins
    Commented Nov 21, 2023 at 4:14

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