+1323 The six faces of a cube are painted black. The cube is then cut into 6^3 smaller cubes, all the same size. One of the small cubes is chosen at random, and rolled. What is the probability that when it lands, the face on the top is black?
+106 Let's first find out how many cubes have how many colors. All cubes must have 0-3 black sides. 8 of the cubes have 3 black spots (all of the corners), 48 have 2 black spots (12 edges each with 4 cubes), and 96 have one black spot (4 by 4 center for each side). If we roll a cube with 3 black sides, it will have a 1/2 chance of landing on a black side, if we roll a cube with 2 black sides, it will have a 1/3 chance of landing on a black side, and if we roll a cube with 1 black side, it will have a 1/6 chance of landing on a black side. Therefore, the total probability is:
\(\frac{8 \times \frac{1}{2} + 48 \times \frac{1}{3} + 96 \times \frac{1}{6}}{216}\)
\( \frac{4+16+16}{216}\)
\(\frac{36}{216}\)
\(\mathbf{\frac{1}{6}}\)
Therefore, you will have a 1/6 chance of rolling a black face.
