# Random Clock Hands

The applet below simulates random selection of one or two clock hands. The dial face is split into red and pink regions whose number depends on the *Sector* settings and whose relative size is determined by the *Ratio*. The event of interest is a hand hitting red.

Obviously, when two hands are selected independently of each other, their probabilities of hitting red must be equal. But what happens if only the first hand is selected randomly while the second hand is drawn at a fixed *Angle* to the first one? How will such an arrangement affect the probabilities for the second hand? (When the *Angle* is zero, only one hand is thrown. Otherwise, there are two of them at the selected *Angle*.)

The string in the low right corner shows the number of red hits for the first (1) and the second (2) hand and the total number of *Trials*.

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Copyright © 1996-2018 Alexander Bogomolny

### 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 *A* be the angle between the two hands. Selecting the second hand is equivalent to selecting a point in *A*, i.e. selecting a random point in *A*,*A*, 2p + *A*). The set **R**_{o} undergoes a shift into **R**_{o} + *A*,*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.

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Copyright © 1996-2018 Alexander Bogomolny

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