Let a,b,c denote the lengths of the sides of our triangle. Without loss of generality, since the triangle is isoceles, let , so that (a,a,b) describes our triangle, where . Now we know additionally , and by the triangle inequality, , So , i.e. , but of course . Thus the possible triangles are described by .
Only one of these is equilateral. Now assuming uniform distribution over these possibilities, and depending whether you include the degenerate (15,15,0), the probability is either or .
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