I tried to script that together after the same idea occurred to me. What's interesting is the selection plane affects the outcome quite heavily. As the plane of random line selection increases in size, the probability of a random chord having length>sqrt3 seems to approach about 0.13

My math might be wrong; doesn't seem to be a floating point precision issue.

        from math import sqrt
        from random import random

        def randomPoints(width):
            return [
                    [random()*width-width/2, random()*width-width/2],
                    [random()*width-width/2, random()*width-width/2]
                ]

        def chordCalc(points):
            def quadraticEquation(a,b,c):
                descriminant=(b**2 - 4*a*c)
                if descriminant <= 0:
                    return None
                x1 = (-b+sqrt(descriminant))/(2*a)
                x2 = (-b-sqrt(descriminant))/(2*a)
                y1 = sqrt(1-x1**2)
                y2 = sqrt(1-x2**2)
                return [[x1, y1],[x2,y2]]

            m = (points[0][1] - points[1][1]) / (points[0][0] - points[1][0])
            b = points[0][1] - m*points[0][0]

            qa = 1+m**2
            qb = 2*b*m
            qc = b**2-1

            return quadraticEquation(qa, qb, qc)

        def distance(points):
            return sqrt((points[0][1] - points[1][1])**2
                      + (points[0][0] - points[1][0])**2)

        chords=0
        largeChords=0
        sqrt3=sqrt(3)
        while True:
            if random() > 0.9999:
                print float(largeChords)/chords
            points = randomPoints(500) #adjust me down to 0.5 and see probability go to ~0.33
            intersections=chordCalc(points)
            if intersections is None:
                continue
            chords += 1
            if distance(intersections) > sqrt3:
                largeChords += 1