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.