riddle of the day

There are geometric probability problems where the exact answer emerges from a surprisingly straightforward argument. This is one of them.

Three thumbtacks on a plate

Strategist
Chance and uncertainty

Riddle statement

Three thumbtacks are placed at random on the edge of a circular plate.

What is the probability that there is some semicircle that contains all three?

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Solution

Take one of the three thumbtacks and consider the semicircle that starts right on it in a clockwise direction.

For that semicircle to contain all three, the other two thumbtacks must fall on that half of the edge. Each one does so with probability 1/2, and both conditions are independent, so the probability is 1/4.

Now, if there is a valid semicircle, it is always possible to rotate it until its end coincides with one of the three thumbtacks. Therefore, the desired event is exactly the union of the three previous cases.

These three cases are disjoint except in configurations of null probability, so that the total probability is:

$3 \times \frac{1}{4} = \frac{3}{4}$