Quote:
Originally Posted by jtwang
How would you determine f(x) == g(x) exactly  since the set of possible hypotheses is infinite (3 reals), wouldn't Pr(f(x) != g(x)) == 1? Obviously you could choose some arbitrary epsilon but then that wouldn't be "exactly."

There are two lines, the original line that determines the separation between +1 and 1, and the line determined by the PLA. The questions ask what fraction of the space is different between the two lines. If they don't cross, that is the area between the two lines (divided by four, the total area). If they do cross, it is the area of two triangles (or, sometimes, quadrilaterals).
Each line can crosses two of the sides of the square. (I suppose it could also go right through a corner, but not likely). Handling all the possible combinations of the two lines is a lot of work.
In another thread I discussed how I did it, only counting, and computing the area of, cases where both lines go through the top and bottom. That is about 30% in my tests. By symmetry, there should also be 30% where both go through the left and right sides of the square The remaining cases might have a little less area, but I based my answer on just the lines going through the top and bottom of the square. Seemed more interesting than the choose random point method.