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 $\mathbf{S}=2$.)