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# Bronx vs brooklyn

## Question

A man lives in Manhattan near a subway express station.  He has two girlfriends, one in Brooklyn, one in the Bronx.  To visit either girl he must take a train.  To go to the one in Brooklyn, he takes a train on the downtown side of the platform, and to visit the one in the Bronx, he takes a train on the uptown side of the same platform.  Since he likes both girls equally well, he simply takes the first train that comes along.  In this way he lets chance determine whether he rides to the Bronx or to Brooklyn.  The young man reaches the subway platform at a random moment each Saturday afternoon.  Brooklyn and Bronx trains arrive at the station equally often--every ten minutes.  Yet for some obscure reason, he finds himself spending most of his time with the girl in Brooklyn.  In fact, on average, he goes there nine times out of ten.  Why are the odds so heavily in favor of Brooklyn?

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Spoiler

The Brooklyn-bound train arrives, each hour, exactly 54 minutes after the Bronx-bound train. The time intervals determine the destination probabilities, and they are in the ratio of 54:6 = 9:1, favoring Brooklyn.

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