Colored Cube: Probability (this is wrong)

A 3 x 3 × 3 cube is composed of 27 1 x 1 × 1 cubes, which are blue by default. All the surfaces of the 3 × 3 × 3 cube are painted green. The cube is then disassembled, and while each small cube may have different orientations, this does not affect the probability of the color of the hidden face. If you select a random small cube and notice that 5 sides visible to you are blue, what is the probability that the last face, which is not visible to you, is green?

Answer

Now to answer, note that we are asked about the ratio of

\frac{\text{Colored cubes with 5 blue, 1 green}}{\text{(all blue + 5 blue, 1 green)}}

This ratio comes from the definition of conditional probability.

Now, only the inner cube has all blue faces—but there is only one such cube, so “all blue” = 1.

Next, a cube with 5 blue and 1 green face must not be on an edge (or vertex), because cubes on edges and vertices have 2 or 3 painted faces, respectively. Therefore, only the non-edge cubes (the center cube of each face) have exactly 1 green face. Since there are 6 faces, there is 1 such cube per face.

Hence, the probability is

P = \frac{6}{6+1} = \frac{6}{7}.

Notes and comments

Comment 1: This is wrong, it should be $\frac{6}{12}$ . I will correct it later.