It's improperly formed as a question - the ruffian can shoot whenever he likes;

Consider:

Does "random" mean

1. uniform distribution on x and y coordinates with some sort of capping at the circle boundary? Or perhaps uniform across all possible x,y pairs inside (on the edge also?) of the circle? what about a normal distribution?

2. a choice of an angle and a length?

3. A point using 1 or 2, and then a random walk for 2 and 3?

I could go on. The worked solution is for random = uniform distribution across all possible reals inside the boundary, I think.

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szczepan1 3 hours ago | parent | next [-]

Author here: when calculating this I _did_ assume a uniform (area) distribution on the unit disk.

Now it does say

> Three points are chosen independently and uniformly at random from the interior of a unit circle.

which sounded OK to me at the time but I understand there could have been some ambiguity. Especially around the "uniform on area" part.

Also, I think that with rejection sampling you could get the same with 1) [0], 2) would work (provided correct scaling) [1]. No idea about 3) or the normal distribution thing you mentioned - I figured the problem was hairy enough already!

[0] https://blog.szczepan.org/blog/monte-carlo/#sampling-uniform... [1] https://blog.szczepan.org/blog/monte-carlo/

	▲	vessenes 3 hours ago \| parent [-]
		Totally agreed! Conceptually I think of the radian/length distribution as having uniformly increasing density closer to the origin - you could imagine a whole bunch of discs concentrically stacked ending in the circumference of the circle - each of them - if a constant "radius length" - will have the same "number" of points but spread out over a larger total area. It's been a lonnnng time since my geometry university courses, but my vague memory is there are some tricky differential geometry historical problems that founder on this precise imprecision. Fun site, thank you for the write up. I skimmed each and every matrix and assumed you did a great job.

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voidmain 3 hours ago | parent | prev [-]

The article currently says

> Three points are chosen independently and uniformly at random from the interior of a unit circle

Has it been edited in the last 15 minutes to address your objection or something?

	▲	szczepan1 3 hours ago \| parent [-]
		Hey, author here :) That has always been the statement (i.e. I've not updated it since adding the post). I do agree that the "uniform on area" bit should have been made more clear!