Random Archery
Robin of Foxley has entered an archery tournament. Her aim is excellent (relatively speaking), as she is guaranteed to hit the circular target. However, her arrows are equally likely to hit each location within the target.
Marian uses a target with 10 concentric circles, whose radii are 1, 2, 3, etc., all the way up to 10 — the radius of the entire target. This time, Robin must fire as many arrows as she can, such that each arrow falls within a smaller concentric circle than the previous arrow.
On average, what score can Robin expect to achieve in this version of the archery challenge?
++