Waiting for a Train

Here's another puzzle. I came across a version of this puzzle in one of Paul Sloane's books, but I don't know who originally created it.

OK, there's a guy called Dennis (why Dennis? Because we're in that sort of mood, that's why. Tell me if this makes you laugh). He has two girlfriends: one lives in City A and the other in City C. Dennis lives in City B, which is between City A and C. He visits one of his girlfriends each weekend. He travels by train: a northbound train takes him to City A; a southbound train to City C.

Now Dennis loves both of these girls equally. So equally, in fact, that he can never decide which one to visit. He leaves his visits in the hands of Fate. When he goes to the station for his weekend away, he arrives at a random time and waits for the first train that comes along: that decides whether he goes north or south. Because both trains leave at half-hour intervals, Dennis figures that he has an equal chance of going north or south. He figures that over the long run, things should even out and he would visit the girlfriends approximately the same number of times.

This is not how things turned out. After a year's worth of visits, Dennis has visited the girlfriend in City A approximately eighty percent of the time. The girlfriend in City C has been sorely neglected.



Anonymous said...

I believe I might have already answered this one for you Fiona, but here's my reasonong anyway:

It is becasue the northbound and southbound train schedules are staggered by 24 minutes, so they would look like this:

North South
12:00 12:24
12:30 12:54
1:00 1:24

This always gives the northbound train a 24-minute time window for Dennis to catch. He only has a 6-minute time window for the southbound train. 24 minutes is 80% of each 1/2 hour interval; 6 minutes is 20% on the same time.


fiona-h said...

quite right :-)

Anonymous said...

I think you've got it flipped. The North column should have the south column's numbers.