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Rob and Steve are having an archery contest: the goal is to shoot an arrow closest to the center of a target (the bull's eye). Let's model the target as R^2; the bull's eye is the origin (0, 0). Suppose that Rob's shot lands at coordinates (Xr, Yr), and the Steve's shot lands at coordinates (Xs, Ys). Suppose that (Xr, Yr) and (Xs, Ys) are uniformly distributed on the disc D where D equals the set {(x,y) | x^2+y^2 <= r^2}.
(i) Rob and the Steve each shoot one time. What is the probability that Rob wins the contest?
(ii) Since Rob tends to be a better shot, the Steve is allowed to shoot twice. What is the probability that the Steve wins? That is, what is the probability that one of the Steve's two shots is closer to the bull's eye than Rob's single shot?
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