13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).

Upon entering the station, the Scientist boards the first train that arrives. It is known that trains run at approximately equal intervals in both directions, and that the train traveling on the northern route (through "Belorusskaya") takes 17 minutes to travel from "Kievskaya" to "Taganskaya" or vice versa, while the train on the southern route (through "Paveletskaya") takes 11 minutes.

Out of habit, the Scientist always calculates everything. Once, he calculated that over many years of observation:

- the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the train traveling clockwise arrives at the same station. The same is true for "Taganskaya";

- the average travel time from home to work is 1 minute less than the travel time from work to home.

Find the expected interval between trains traveling in the same direction.

![](https://cdn.mathpix.com/cropped/2024_05_06_ca2fb3a020271aee0b0ag-08.jpg?height=785&width=785&top_left_y=1606&top_left_x=1138)
Solution. If the Scientist boarded trains of different directions with equal probabilities, the average travel time in one direction and the average travel time in the other would be the same. Therefore, the probabilities are not equal.

Let $p$ be the probability that the Scientist boards a train going clockwise. Then the expected travel time from "Taganskaya" to "Kievskaya" is

$$
11 p + 17(1-p) = 17 - 6p
$$

On the return trip from "Kievskaya" to "Taganskaya," the expected travel time is

$$
17 p + 11(1-p) = 11 + 6p
$$

According to the condition, $11 + 6p - (17 - 6p) = 1$, from which $p = \frac{7}{12}$. Let the interval between trains be $T$. Then $T(1-p) = Y$, where $Y$ is the time between the arrival of a train "clockwise" and the arrival of a train "counterclockwise" at the favorite stations. Then

$$
\mathrm{E} T = \frac{\mathrm{E} Y}{1-p} = \frac{5}{4} \cdot \frac{12}{5} = 3
$$

Answer: 3 minutes.