7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, the probability that a car will turn off the highway onto each of these roads is $\frac{1}{n}$. The road by which a car leaves the highway is independent of the road by which it entered the highway.

![](https://cdn.mathpix.com/cropped/2024_05_06_18848d423dc3e63c21dfg-05.jpg?height=411&width=532&top_left_y=654&top_left_x=1393)
which the car leaves the highway, is independent of the road by which it entered the highway.

Find the probability that a random car that has entered the highway will pass the $k$-th intersection (either drive through it or turn at it).
Answer: $\frac{2 k n-2 k^{2}+2 k-1}{n^{2}}$.

Solution. The specified event will occur in one of three cases.