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, which has no subdivisions - it's just one big circle. However, her arrows are equally likely to hit each location within the target.
Her true love, Marian, has issued a challenge. Robin must fire as many arrows as she can, such that each arrow is closer to the center of the target than the previous arrow. For example, if Robin fires three arrows, each closer to the center than the previous, but the fourth arrow is farther than the third, then she is done with the challenge and her score is four.
On average, what score can Robin expect to achieve in this archery challenge?
++