Is there a formal proof that points taken at random in a bound area are evenly distributed?

I am an amateur trying to understand how probability works on the euclidean plane.

Despite my efforts I couldn't find any formal proof that points taken at random in a bound area are evenly distributed. I.e. if we have an area split in two equal halves and choose a number of points at random in this area, the number of points in each half is expected to be half the number of the total points, (no matter how the original split was made).

So my simple question is: is there such a proof?

In other words: does the euclidean plane dictate the even distribution, or we decide what the distribution is, each and every time we take points at random?

• "taken at random" without any other qualifications implies uniformly at random.
– qwr
Commented Jun 26 at 0:18

• I don't agree that "points taken at random from some set" implies that (a) the points follow a uniform distribution relative to your favourite measure, or that (b) several points are independent. To me, the correct formulation should be something like"Let $P_1, P_2, \ldots, P_n$ be random points in $X$, independently indentically distributed uniformly by area". Of course, the main issue remains the same: the points are evenly distributed by definition because we say so. Commented Jun 25 at 22:25