A 3x3x4 cuboid is painted blue and cut into 1x1 cubes. what is the expected value for the painted sides of a randomly selected cube?

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Mathematics

Finnegan Landry

30 December, 17:25

A 3x3x4 cuboid is painted blue and cut into 1x1 cubes. what is the expected value for the painted sides of a randomly selected cube?

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Benson · Answer 1

A 3 x 3 x 4 cuboid can be cut into 3 (3) (4) = 36 pieces of 1 x 1 cubes.

Out of the 36 pieces of 1 x 1 cubes, there are no 1 x 1 cube with the four sides painted, there are eight 1 x 1 cubes with simply 3 sides painted, there are twenty 1 x 1 cubes with only 2 sides tinted, there are 6 cubes with only 1 side painted and there are 2 cubes with no side painted.

The chance that the four sides of a randomly selected cube is painted is 0.

The likelihood that only 3 sides of a randomly selected cube is painted is 8 / 36 = 2 / 9.

The chance that only 2 sides of a randomly selected cube is painted is 20 / 36 = 5 / 9.

The likelihood that only 1 side of a randomly selected cube is painted is 6 / 36 = 1 / 6.

The probability that no side of a randomly selected cube is painted is 2 / 36 = 1 / 18.

Thus, the expected value for the painted sides of a randomly selected cube is given by 4 (0) + 3 (2 / 9) + 2 (5 / 9) + 1 (1 / 6) + 0 (1 / 18) = 2 / 3 + 10 / 9 + 1 / 6 = 35 / 18 = 1.94