The two-dimensional version of this question was already asked (although in a different language) here.

In fact, besides the generalization to measurable sets mentioned by Pete, this result can be generalized in the following way. A set of $k$ points $x_1,\ldots,x_k$ in $\mathbb{Z}^n$ is said to be primitive if the open paralellotope $\left\{ \alpha_1 x_1 + \ldots \alpha_k x_k : 0 < \alpha_i < 1 \right\}$ contains no integer point. The "probability" that a set of $d$ points is primitive equals to $$\frac{1}{\zeta(n) \zeta(n-1) \ldots \zeta(n-k+1)},$$ as shown, for example, in this paper.

One way of formalizing the notion of "probability" is to define it for the restriction of $\mathbb{Z}^n$ to some ball of radius $R$ and let $R \to \infty$, as done in this paper . This is sometimes called the "natural density" of $\mathbb{Z}^n$.

The two-dimensional version of this question was already asked (although in a different language) here.

In fact, besides the generalization to measurable sets mentioned by Pete, this result can be generalized in the following way. A set of $k$ points $x_1,\ldots,x_k$ in $\mathbb{Z}^n$ is said to be primitive if the open paralellotope $\left\{ \alpha_1 x_1 + \ldots \alpha_k x_k : 0 < \alpha_i < 1 \right\}$ contains no integer point. The "probability" that a set of $d$ points is primitive equals to $$\frac{1}{\zeta(n) \zeta(n-1) \ldots \zeta(n-k+1)},$$ as shown, for example, in this paper.

One way of formalizing the notion of "probability" is to define it for the restriction of $\mathbb{Z}^n$ to some ball of radius $R$ and let $R \to \infty$, as done in this paper . This is sometimes called the "natural density" of $\mathbb{Z}^n$.