The two-dimensional version of this question was already asked (although in a different language) [here](http://math.stackexchange.com/questions/10998/probability-that-a-vector-in-mathbbzn-is-primitive).

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](http://www.sciencedirect.com/science/article/pii/S0022314X06002733).

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](http://arxiv.org/pdf/1005.3967.pdf) . This is sometimes called the "natural density" of $\mathbb{Z}^n$.