[ The ratio of the areas of triangles with a common base or common height ]

In one island tribe, there is a custom - during a ritual dance, the shaman tosses three thin straight rods of the same length, tied in a shape resembling the letter П, high into the air. Adjacent rods are connected by a short thread and can therefore rotate freely relative to each other. The rods fall onto the sand, forming a random figure. If a self-intersection occurs (the first and third rods cross each other), the tribe will face poor harvests and various troubles in the coming year. If there is no self-intersection, the year will be successful - bountiful and happy. Find the probability that the rods will predict good fortune for the year 2017.
A broken line formed by rods is called $A B C D$. Let $\alpha=\angle A B C$ be the angle between the first and second links, and $\beta=\angle B C D$ be the angle between the second and third (see the left figure). We can assume that $0 \leq \alpha \leq \pi$ and then $0 \leq \beta \leq 2 \pi$. The elementary outcomes are pairs $(\alpha, \beta)$. On the coordinate plane $\alpha O \beta$, they fill a rectangle $G$ (see the right figure), with the probability of a point $(\alpha, \beta)$ falling inside a certain figure being proportional to the area of that figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_864d5071b377f8fb364bg-03.jpg?height=488&width=1288&top_left_y=1086&top_left_x=386)

The first and third links $A B$ and $C D$ intersect at some point $K$ only if the ray $C D$ lies inside the angle $B C A$, and the ray $B A$ lies inside the angle $C B D$, i.e., $0 \leq \angle B C D < \angle B C A$, and $0 \leq \angle A B C < \angle C B D$.

Considering that triangles $A B C$ and $C B D$ are isosceles, we get the conditions $0 \leq \beta < \pi / 2 - \alpha / 2$ and $0 \leq \alpha < \pi / 2 - \beta / 2$. On the coordinate plane $\alpha O \beta$, these inequalities define a quadrilateral $F$ inside the rectangle $G$. The probability of self-intersection

$P(F) = S_{F} / S_{G} = 1 / 12$. Therefore, the desired probability is $1 - 1 / 12 = 11 / 12$.

## Answer

$11 / 12$.