I found an article in the American Math Monthly, vol 103 #4 p.308, which discusses the problem and gives a number of references.
The probability that three points picked uniformly and independently from a disk form an obtuse triangle is 9/8-4/pi^2=.7187.... It's been conjectured that this is the shape which gives the lowest probability of obtuseness if you pick three points randomly and uniformly from it. (As usual, one can take any radius, or let the radius go to infinity, without changing the answer.) A Gaussian distribution on the plane, as has been noted, gives the answer 3/4=.75.
The probability of obtuseness of the triangle formed by three points taken at random from an n-ball tends toward zero as n goes to infinity. Such a triangle becomes ever more likely to approximate an equilateral triangle.
The problem of determining the probability of obtuseness of the triangle formed by three points picked at random from a plane (without specifying what that's supposed to mean) was included in C.L.Dodgson's (Lewis Carroll's) "Pillow Problems", problems he suggested thinking about before going to sleep.
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