I stand corrected. Thank You. That's what happens when you are in too big of a hurry. You are also correct in that there is more to this problem than meets the eye. Thanks for the update and I will soldier on - Ian

I also want to suggest harder problem which I am still trying to solve. Go one dimension higher and now consider two concentric circles instead of two pairs of points. Find the probability for a random section inside the big circle to cross the smaller circle.

Choose 2 points P1 and P2 at random inside the large circle. What is the probability that the line joining P1 and P2 will cross the smaller circle. I will show 2 methods.

1) P1 is in the small circle. P = kr^2 / kR^2 = r^2 / R^2

P2 must lie in the ring. P = k(R^2-r^2) / kR^2 = (R^2 - r^2)/ R^2

Mulitply since both must happen

P(crossing) = r^2(R^2 - r^2) / R^4

We repeat the process for P2 in the small circle and P1 in the ring and get the same answer
so we double.

P(crossing)

= [r^2/R^2] [(R^2-r^2)/R^2] *2

= 2r^2(R^2 - r^2) / R^4

2) P(crossing)

= 1 - both lie in the ring - both lie in the small circle

@Ian Fowler I understand your analysis, but it's not correct. If P1 and P2 both lie in the ring, they still can cross the smaller circle (they can also touch it):

Let φ=r/R. Then the line joining P1(x1,y1) and P2(x2,y2)

1) Definately never crosses if they both lie in the small circle

Probability of that happening = φ^4

2) Definately crosses if they lie in different areas (you considered it would be enough)

P = 2φ^2(1-φ^2)

3) We don't know/depend on the angle/distance from the centre/a lot of parameters

Check this out.

I stand corrected. Thank You. That's what happens when you are in too big of a hurry. You are also correct in that there is more to this problem than meets the eye. Thanks for the update and I will soldier on - Ian

I also want to suggest harder problem which I am still trying to solve. Go one dimension higher and now consider two concentric circles instead of two pairs of points. Find the probability for a random section inside the big circle to cross the smaller circle.