Consider a projective space $\mathbb{P}^n(k)$ of dimension $n > 0$ over an infinite field $k$. Let $\Delta$ be a hyperplane in $\mathbb{P}^n(k)$. Assume that we chose a point $P$ "at random" (we suppose that chosing points happens with equal probability).
What is the best natural way to express what the probability is that $P$ is contained in the affine space $\mathbb{P}^n(k) \setminus \Delta$ ? (With "point," I mean $k$-rational point.)
The probability tends to 1, but how does one express this in a precise way ? (Through measures ?)
$\newcommand{\PP}{\mathbb{P}}\newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}}$ Choosing points "at random" isn't always well defined on $\PP^n(k)$ for all $k$; for instance, in the case of $k = \QQ$, $\PP^n(\QQ)$ is countable and thus there can be no uniform distribution.
If $k$ is a finite field, then the probability need not tend to $1$ if you use the counting measure. Write $\PP^n(k)$ in coordinates $(x_0,x_1,\ldots,x_n)$. Then the probability you hit the hyperplane defined by $x_0 = 0$ is precisely $1/q$, which doesn't tend to zero. More generally, the probability that you hit a codimension $j$ hyperplane will be $1/q^j$.
In the case of $\RR$ or $\CC$ you can't choose something uniformly, but can you look at the induced Lebesgue measure on projective space and indeed get that the induced measure of a hyperplane is zero.