The Unfair Subway ----------------- (adapted from "Fifty Challenging Problems in Probability with Solutions", by F. Mosteller. Dover Press.) Problem Phil used to study in Cambridge, Massachusetts. After work (some random time in the middle of the night) Phil would walk out of his building and go to the subway, where he'd board the first train, whether it was going North or South. This worked fine because his girlfriend's house was to the North, and his own home was to the South. Both the Northbound and the Southbound trains run every 10 minutes. Phil observes that, over the course of several months, he finds himself going home only about 3 times per month. What is going on? And on the average, how long does Phil wait for a train? You may assume that Phil arrives at a random time and that he goes to work every day. Solution Suppose that the Northbound trains leave at 8:00 8:10 8:20 8:30 etc. while the Southbound trains leave at 8:01 8:11 8:21 8:31 etc. Then in each 10 minute interval there are 9 minutes such that Phil will go North, and 1 minute such that Phil will head south. Thus 10% of the time Phil will go South, which is consistent with going home about 3 days a month. The expected waiting time is (.1)(.5) + (.9)(4.5) = 4.25 mins, since 1/10 of the time there will be an expected wait of 1/2 minute, and 9/10 of the time there will be an expected wait of 4.5 minutes.