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A fact noticed by physicist G. Gamow 
when he had an office on the second floor and physicist
M. Stern had an office on the sixth floor of a seven-story building (Gamow and Stern 1958, Gardner
1986). Gamow noticed that about 5/6 of the time, the first elevator to stop on his floor was going down, whereas about the
same fraction of time, the first elevator to stop on the sixth floor was going up. This actually makes perfect sense, since 5
of the 6 floors 1, 3, 4, 5, 6, 7 are above the second, and 5 of the 6 floors 1, 2, 3, 4, 5, 7 are below the sixth. However,
the situation takes some unexpected turns if more than one elevator is involved, as discussed by Gardner (1986).
References
Gamow, G. and Stern, M. Puzzle Math. New York: Viking, 1958.
Gardner, M. ``Elevators.'' Ch. 10 in Knotted Doughnuts and Other Mathematical Entertainments.
New York: W. H. Freeman, pp. 123-132, 1986.