Three thumbtacks are placed randomly on the edge of a circular plate. What is the probability that there is some semicircle that contains all three?
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Three chinchetas in a plato
It belongs to that delightful family of probabilities on the circle where a seemingly chaotic situation ends up governed by a very simple figure.
Hints
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- If a semicircle works, you can rotate it until one end matches one of the thumbtacks.
- Look at a specific pin and ask yourself the probability that the other two will fall within the semicircle that begins with it.
- That case has probability 1/4, and there are three possible thumbtacks that can act as the initial edge.
Solution
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Take one of the three thumbtacks and suppose that the semicircle we are looking for begins right on it. For that semicircle to contain all three, the other two thumbtacks must fall within the corresponding half of the border.
Each one falls there with probability 1/2. So, for a fixed thumbtack, the probability of it working is 1/4.
Now, the valid semicircle, if it exists, can start on any of the three pushpins. And these cases are disjoint except for ties of zero probability.
Therefore, the total probability is 3 × 1/4 = 3/4.
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