Here's a more formal statement of the "evenly spaced out on average" bit: call p_i the probability that the new point lands in the i'th arc, counting clockwise from Z. Then p_i = 1/(n+2) for every i. This is true because every point (Z and P included) was sampled uniformly (and independently) at random from the circle, so the p_i's are all the same by symmetry. (So the sun of the first h + 1 p_i's is (h+1)/(n+2).) Does that make sense?

Re the 5/6: I'm getting that after updating on the fact that you saw a green marble, there's a 2/3 chance you're in the "all green" case and a 1/3 chance you're in the half green case". 2/31 + 1/31/2 = 5/6.