355.The Hidden Face
★★★conditional probabilityHudson River Trading
A 3x3x3 cube is painted red on its entire outer surface and cut into 27 unit cubes. One piece is chosen at random and placed on a table in a random orientation, so that five of its faces are visible and all five turn out to be unpainted. What is the probability that the hidden sixth face is red?
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Only two kinds of piece can show five clean faces. Count the orientations, not just the pieces.
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