# Average Waiting Time

At Berracan station, northbound trains arrive every 3 minutes. starting at noon and finishing at midnight. Each Day i walk to the Berrabcan station at a random time in the afternoon and wait for the first train in either direction, on average how many second should i expect to wait?

Note by Yasir Soltani
2 years, 4 months ago

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As Chew mentioned, your problem is currently incompletely stated.

The scenario of 1 train is dealt with by Chew.

For the scenario of 2 trains, in a similar manner, draw the graph of waiting time against arrival time. Then, find the expected value of it (through integration, or finding areas). Depending on how the train arrives, the answer would range from 0.75 to 1.5.

Staff - 2 years, 4 months ago

I am not so good in combinatorics. Do check my answer.

You mention about northbound trains arrive every 3 minutes but no mention about southbound trains. But you mention about the first train in either direction. If only considering one direction then the expected waiting time should be:

$$$$\begin{split} E[T] & = \int_0^3 t f(t) dt \quad \quad \small \color{blue}{\text{where } t \text{ is the waiting time and } f(t) \text{ is the probability distribution function of }t} \\ & = \int_0^3 \frac{t}{3} dt \\ & = \left[\frac{t^2}{6}\right]_0^3 \\ & = \frac{3}{2} \text{ minutes}\end{split}$$$$

Expecting time of $$\frac{3}{2}$$ minutes (half the arrival time of the trains) is obvious. If it is for two directions, it is much more complicated and I don't have a solution.

- 2 years, 4 months ago

- 2 years, 4 months ago