Uniform distributions and random triangles

Abstract

“What is the probability that a randomly drawn triangle is acute-angled?” This question has been discussed in recent Gazettes (for example by Trevor Easingwood in [2], followed up by Stephen Ainley in [1]). As Easingwood pointed out, the answer to this question depends on what properties of triangles are assumed to be random and on what one means by random. And as Ainley pointed out some of these assumptions can lead to a contradiction. A statistical model is necessary to define the sense in which the chosen properties (lengths of sides, angles, co-ordinates of vertices,…) are assumed random. This model may (and preferably will) relate to a possible physical ‘random’ mechanism for generating the random triangle.