Random Clock Hands

Problem

random clock hands, problem

Discussion

As hard as it may be to believe, the events of the two hands hitting red is independent even when they are 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 $\textbf{R}$ 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 $\textbf{R}_{o}$ in the interval $[0, 2\pi).$ Let $\alpha$ be the angle between the two hands. Selecting the second hand is equivalent to selecting a point in $[0, 2\pi)$ and then shifting it by $\alpha,$ i.e. selecting a random point in $[0, 2\pi) + \alpha,$ or which is the same $[\alpha, 2\pi + \alpha).$ The set $\textbf{R}_{o}$ undergoes a shift into $\textbf{R}_{o} + \alpha,$ which has the same relative size as the set $\textbf{R}_{o}.$ Furthermore, the original set $\textbf{R}$ 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.

Thus the answer two all three questions is $1.$

 

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