342.Corner of a Painted Cube
★★★conditional probabilityJane Street
A 3x3x3 cube is painted on all six outer surfaces and then cut into 27 unit cubes. One unit cube is drawn at random and placed in front of you with a random face towards you, and that visible face turns out to be painted. What is the probability that the piece is a corner piece?
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Condition on the face you are looking at rather than on the cube. Corner pieces offer three painted faces each, so they are over-represented among painted faces.
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