I have a really cool maths problem for you! Pick two random points on the arc of a circle. What's the probability that the straight line that connects these two points is longer than the radius of the circle?

I'm still working on this problem myself. I've been toying around with a lot of different ideas and I've learnt quite a bit about the circle in the process. One thing I tried:

upper semi-circle - lower semi-circle = radius (r)
sqrt(r^2 - x^2) + sqrt(r^2 - x^2) = r
solving for r we get: r = +- 2x/sqrt(3)

if we solve for x we can obtain the values of x for which the upper semi-circle - lower semi-circle = r

therefore, solving for x we get: x = +- r * sqrt(3) / 2

therefore, - r * sqrt(3) / 2 < x < r * sqrt(3) / 2 then sqrt(r^2 - x^2) - -sqrt(r^2 - x^2) > r

now, the number of chords in a circle > radius of the circle (provided the chords are all parallel to the y-axis) = r * sqrt(3) / 2 - -r * sqrt(3) / 2

let the number of chords = n

2 * r * sqrt(3) / 2 = n

r * sqrt(3) = n

that is to say, the number of chords in a circle > radius of the circle = r * sqrt(3) provided the chords are all parallel to the y-axis.

I thought that was really interesting.

Now, the probability of an event is defined as the number of outcomes in the event n(E) over the number of outcomes in the sample space n(S). I've managed to determine the number of outcomes in the sample space for this event... it's simply the circumference of the circle * the circumference of the circle, or 4 * pi^2 * r^2. The remaining piece of the puzzle is what n(E) is. I think the answer lies somewhere in the research I've done... I just have to determine the TOTAL number of chords in a circle > the radius of the circle (i.e. the chords are not limited to being parallel to the y-axis). Then you divide that by n(S) = 4 * pi^2 * r^2 and voila! I'm guessing that n(E) contains the expression r^2 somewhere, so the radius cancels out. (Otherwise P(E) - the probability of the event - will contain the variable r, which is obviously undesirable... we want just a number without any variables.)

I'm going to take an educated guess and say that the TOTAL number of chords in a circle > radius of the circle = r * sqrt(3) * 1/2 * circumference of the circle (the reason that we multiply by only HALF the circumference of the circle is that the second half of the circle contains exactly the same chords as the first half... in actual fact, we should multiply by 1/2 * (circumference of the circle - 1/infinity) because the chords at the highest point of the circle are the same as the chords on the lowest part of the circle, but 1/infinity tends to zero so leaving it out of our calculations is still accurate enough for all practical purposes) = r * sqrt(3) * pi * r = sqrt(3) * pi * r^2

If that is true, then

P(E) = n(E) / n(S)
     = (sqrt(3) * pi * r^2) / (4 * pi^2 * r^2)
     = sqrt(3) / (4 * pi)
     = 1,732 / 12.566
     = 0.138

But I don't know if this is right or not. I guess I could write a program to test this out.

All in all, quite an interesting (and frustrating) puzzle! Good luck with it! (This somewhat long rant will probably help get you started out, although I wouldn't be surprised if your method of approach differs entirely from mine. Let me know what you come up with!)