All faces of a wooden cube with an edge length of $10\, \mathrm{cm}$ were painted blue. After drying, the cube was cut into $1000$ equally-sized little cubes with an edge length of $1\, \mathrm{cm}$ each. What is the probability that when randomly selecting one of these little cubes, we draw the one that has exactly two blue faces?
Martina solved the problem as follows:
(1) The wooden cube has a total of $12$ edges.
(2) Each edge separates $2$ of its blue faces.
(3) On each edge, there are $10$ such little cubes that have exactly $2$ blue faces, e. g. $120$ little cubes have exactly two blue faces.
(4) The probability of drawing a little cube with two blue faces out of $1000$ is $\frac{120}{1000} = 0.12$.
Martina made a mistake in step (3). What is the mistake?
The little cubes that contain the vertices of the original cube have three blue faces. Therefore, on each edge, there are only $8$ little cubes with two blue faces. By cutting, there were created a total of $96$ $(=8\cdot12)$ cubes with two blue faces.
The little cubes that contain the vertices of the original cube are counted twice. Therefore, the number of little cubes that have exactly two blue faces is $112$ $(=120-8)$.
The little cubes that contain the vertices of the original cube are counted three times. Therefore, the total number of little cubes that have exactly two blue faces is $104$ $(=120 – 2\cdot8)$.
The little cubes that contain the vertices of the original cube have three blue faces. Therefore, there are $8$ little cubes with two blue faces on each edge, but they are counted twice. The number of little cubes with two blue faces is $48$ $\left(=\frac{8\cdot12}{2}\right)$.
(1) The wooden cube has a total of $12$ edges.
(2) Each edge separates $2$ of its blue faces.
(3) There are $10$ cubes on each edge. However, those that contain the vertices of the original cube have three blue faces. Thus, there are $8$ cubes left on each edge that have exactly $2$ blue faces, i.e., $96$ cubes have exactly two blue faces.
(4) The probability of drawing a little cube with two blue faces out of $1000$ is $\frac{96}{1000} = 0.096$.