Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set out on a journey from their cabin at a speed of 4 km/h along a road each arbitrarily chooses. Determine the probability that the distance between them after one hour will be at least 6 km.

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# Solution.

Let's find the distance between the geologists after 1 hour if they are walking on adjacent roads (Fig. 4).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-08.jpg?height=326&width=343&top_left_y=1145&top_left_x=868)

$60^{\circ}$

Fig. 4

Since the triangle is equilateral, $x=4$, which is less than 6.

Let's find the distance between the geologists after 1 hour if they are walking on roads that are one apart (Fig. 5).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-08.jpg?height=411&width=443&top_left_y=1733&top_left_x=818)

Fig. 5

$\frac{y}{2}=4 \sin 60^{\circ}=\frac{4 \sqrt{3}}{2}=2 \sqrt{3}, y=4 \sqrt{3}$, which is not less than 6.

Thus, the distance will be no less than 6 km if the geologists do not choose the same or adjacent roads.

Possible outcomes. The first geologist can choose any of the 6 roads, and the second geologist can also choose any of the 6 roads, i.e., $n=6 \cdot 6=36$.

Favorable outcomes. The first geologist can choose any of the 6 roads, and the second geologist can choose only 3 (not the same or adjacent roads), i.e., $n=6 \cdot 3=18$.

Then, $P=\frac{18}{36}=0.5$.

Answer. $0.5$.