Grade 10, waiting for the train.

Question

At Berracan station, northbound trains arrive every three minutes starting at noon and finishing at midnight while southbound trains arrive every five minutes starting at noon and finishing at midnight. Each day, I walk to Berracan Station at a random time in the afternoon and wait for the first train in either direction. On average, how many seconds should I expect to wait.

The thing i dont understand is that if she walks in to the station at, lets say 12:03, would she have to wait for 0 seconds (assuming that she catches the train) or would she have to wait for 2 min (assuming that she misses the train and has to wait for the one at 12:05)?

Answer 1

In $15$ minutes, here is the pattern of number of minutes between arrivals: $3,2,1,3,1,2,3$. Hence the average times for each distance are respectively $3/2,1,1/2,3/2,1/2,1,3/2$. Performing a weighed mean: $$\frac{2\times(3\times(3/2)+2\times 1 +1/2) + 3\times(3/2)}{15}=\frac{37}{30} \text{ minutes}=74 \text{ seconds}$$

Answer 2

The chance of arriving precisely when a train arrives is zero, so you don't care. If she arrives just after the noon train she would wait 3 minutes for the 12:03 northbound. You have a pattern that repeats every $\operatorname{LCM}(3,5)=15$ minutes. Plot out the time waiting from $12:00$ to $12:15$, compute the area under the curve, and divide by $15$ minutes.