The Mother and the Girlfriend

John gets off work at random times between 3 and 5 PM. His mother lives uptown, his girlfriend lives downtown. John takes the first subway that comes in either direction and eats dinner with the one he is delivered to. Even though John believes he has 50-50 chance to have dinner with either his mother or his girlfriend, he visited the former only 2 times out of the last 20. How come?

The subway heading downtown arrives at 3:00, 3:10, 3:20, etc, and the subway heading uptown arrives at 3:01, 3:11, 3:21, etc. Thus, the chance that John goes to his girlfriend is about 90% (depending on train delays).

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  1. John’s assertion that his chances are 50-50 are misled, as he is assuming that during rush-hour (3:00 PM to 5:00 PM), there are equal number of trains travelling in both directions, but that is incorrect, because during rush-hour, one would expect the majority of trains travel uptown to go back home to the suburbs, and not downtown to his city slicker girlfriend’s house.