Discussion

As hard as it may be to believe, the probability of the second hand hitting red is independent of the fact that the two hands is rigidly attached at a certain angle. Imagine throwing the first hand randomly, attaching the second hand at a certain angle, but only drawing the latter. Would you think that the second hand (which actually appears alone) is less random or has a different probability of hitting red than the lone first hand?

To add a little more rigor to the above argument, let R be the set of red points on the unit circle. (Selecting a hand is equivalent to picking a point on the unit circle.) We may also think of it as a set R_o in the interval [0, 2p). Let A be the angle between the two hands. Selecting the second hand is equivalent to selecting a point in [0, 2p) and then shifting it by A, i.e. selecting a random point in [0, 2p) + A, or which is the same [A, 2p + A). The set R_o undergoes a shift into R_o + A, which has the same relative size as the set R_o. Furthermore, the original set R is clearly left unchanged by the shift. What does change is the point at which the circle is cut before being unfolded into a segment.