Let's say a train arrives at a stop in intervals of 15 or 45 minutes, each with equal probability 1/2 (so every time a train arrives, it will randomly be either 15 or 45 minutes until the next arrival). What is the expected waiting time of a passenger for the next train if this passenger arrives at the stop at any random time. This means that the passenger has no sense of time nor know when the last train left and could enter the station at any point within the interval of 2 consecutive trains.
I was told 15 minutes was the wrong answer and my machine simulated answer is 18.75 minutes. I just don't know the mathematical approach for this problem and of course the exact true answer. Sincerely hope you guys can help me. Thanks!
Best Answer
Your simulator is correct. Since 15 minutes and 45 minutes intervals are equally likely, you end up in a 15 minute interval in 25% of the time and in a 45 minute interval in 75% of the time.
In a 15 minute interval, you have to wait $15 \cdot \frac12 = 7.5$ minutes on average.
In a 45 minute interval, you have to wait $45 \cdot \frac12 = 22.5$ minutes on average.
This gives a expected waiting time of $$\frac14 \cdot 7.5 + \frac34 \cdot 22.5 = 18.75$$