Solution: The Inspection Paradox – in a nutshell.

Exercise

You arrive at someone’s home and are ushered into the garden. You know that a train passes the end of the garden every half an hour on average but the trains are actually scheduled so that half pass by with an  interval of a quarter of an hour and half with an interval of 45 minutes. Given that you have no clue when the last train passed by and the scheduled interval between that train and the next, how long can you expect to wait for the next train?

Solution to Exercise

The mean interval between trains is 30 minutes, so the average expected wait would seem to be 15 minutes if you arrive at a random time.

But it is three times as likely that you will arrive during the 45 minutes interval as during the 15 minutes interval, and therefore three times the chance of waiting 22.5 minutes (half way along the 45 minutes interval) as 7.5 minutes (half way along the 15 minutes interval).

So your expected wait is 3 x 22.5 minutes plus 1 x 7.5 minutes, divided by four. This equals 75 divided by 4 or 18.75 minutes (18 minutes, 45 seconds).