Question

Alice arrives at a train station at a time uniformly distributed between 8 p.m. and 9 p.m., and boards the first train that arrives. There are two types of trains that arrive between 7 a.m. and midnight: those that head to town A arrive every 15 minutes starting at 7 a.m., and those that head to town B arrive every 15 minutes starting at 7:05 a.m..

a. How long does Alice wait, on the average?
b. What proportion of days does she go to A?

Answer 1

a) She waits in average 4 minutes and 10 seconds.

b) She goes to A 2/3 of the days

Explanation:

The trains reach the station at 8 p.m, 8:05, 8:15, 8:20, 8:30, 8:35, 8:45, 8:50 and 9 p.m between 8 and 9 pm.

Lets analyze how much does Allice waits in average in each 5 minutes interval:

between 8 and 8:05: She waits in average 2.5 minutes

between 8:05 and 8:10: she waits in average 7.5 minutes (she will have to wait at least 5 minutes and any amount uniformly between 5 and 10).

between 8:10 and 8:15: in average she waits 2.5 minutes.

Other intervals where she waits 2.5 minutes in average: 8:15-8:20, 8:25 - 8:30, 8:30 - 8:35, 8:40-8:45, 8:45-8:50 and 8:55 and 9.

Other intervals where she waits in average 7.5 minutes: 8:20-8:25, 8:35-8:40, 8:50-8:55

There are a total of 4 intervals where she waits in average 7.5 minutes and 8 intervals where she waits in average 2.5 minutes. Taking the average over the intervals we have that she waits in average

(8*2.5 + 4*7.5)/12 = 4.16667 minutes = 4 minutes and 10 seconds

b) She goes to A whenever she arrives in the following 10-minute periods:

8:05-8:15, 8:20-8:30, 8:35-8:45, 8:50-9

So, out of the 60 minute period where she can arrive uniformly, in 40 of those 60 minutes she goes to A, thus she goes to A with a proportion of 40/60 = 2/3.