Jack decides to draw a triangle as follows: he draws a fixed line and draws two more lines which intersect it at distinct fixed points, at an angle chosen uniformly at random over the interval [0,pi].
1) What is the probability that Jack draws a non-degenerate, non-isosceles, non-right-angled triangle?
However, like most people Jack will tend to interpret what he sees to make it conform to a particular pattern. In Jack's opinion there are four kinds of 'special' triangle - right-angled, isosceles, equilateral and right-angled isosceles. Jack may 'interpret' the angles in his triangle by up to x radians (ie he sees each angle as taking some value within x radians of its true value, though his 'interpreted' angles must add up to pi) in order to see a special triangle, and he looks for each kind of special triangle individually. Ignore the possibility of Jack seeing degenerate triangles or triangles which contain two right angles.
2) Let A be the event that Jack sees a right-angled triangle, B isosceles, C equilateral, D r-a isosceles. Write down the probabilities of all 16 possible combinations of these events (eg A and B but not C or D) in terms of x (or if you prefer, use degrees by means of y = 180x/pi).
3) What is the probability that Jack sees a combination of events (eg C and D) that is impossible for any given fixed triangle?