How common are solutions to equations?

Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.

I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.