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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?

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

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The definition of "points taken at random in a bound area" is that the number of points in a subregion is proportional to its area. So there's nothing to prove.

If you have an algorithm that claims to choose random points that way then you have to prove your algorithm is correct. You can test it in practice by dividing the area in half and counting the number of points that end up in each half. There are statistical methods that will tell you how close to equal those counts should be.

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    $\begingroup$ 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. $\endgroup$
    – Hagen von Eitzen
    Commented Jun 25 at 22:25
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    $\begingroup$ @HagenvonEitzen In this question I think independence and area measure are the natural (implicit) assumptions. $\endgroup$
    – Ethan Bolker
    Commented Jun 25 at 23:52

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