5. A race on an unpredictable distance is held as follows. On a circular running track, two points $A$ and $B$ are randomly selected (using a spinning arrow), after which the athletes run from $A$ to $B$ along the shorter arc. A spectator has bought a ticket to the stadium and wants the athletes to run past his seat (then he will be able to take a successful photo). What is the probability that this will happen?

(A. A. Tesler)
Solution. Identify each point on the path with its distance to the viewer clockwise. Then pairs $(A, B)$ can be identified with pairs of numbers from $[0,1)$ (when measured in kilometers). In this case, the probability that $(A, B)$ belongs to some subset of $[0,1) \times[0,1)$ is equal to the area of this subset. We are interested in the set of $(A, B)$ such that $|A-B|>\frac{1}{2}$ (in this case, the shortest arc passes through $0$), which is a pair of triangles with a total area of $\frac{1}{4}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c57db349868d710956b0g-2.jpg?height=309&width=303&top_left_y=802&top_left_x=1613)

Answer: $\frac{1}{4}$.