Question

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?

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 only 3 sides painted, there are twenty 1 x 1 cubes with only 2 sides painted, there are 6 cubes with only 1 side painted and there are 2 cubes with no side painted.

The probability that the four sides of a randomly selected cube is painted is 0. The probability that only 3 sides of a randomly selected cube is painted is 8 / 36 = 2 / 9. The probability that only 2 sides of a randomly selected cube is painted is 20 / 36 = 5 / 9. The probability 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.

Therefore, 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