Question

A cube has all of its sides painted yellow and then is cut into 729 cubes of the same size. All cubes are placed in an urn and are thoroughly mixed, so that the probability of being randomly picked from the urn is the same for all cubes. What is the probability that a randomly picked cube has exactly two of its sides painted yellow

Answer 1

Here there is a cube whose all of its sids are painted yellow/ Ow its cut into 729 cubes that means 9³ cubes of equal sizes.

All cubes are placed in an urn and are thoroughly mixed, so that the probability of being randomly picked from the urn is the same for all cubes.

Now the we have to find the probability that  a randomly picked cube has exactly two of its sides painted yellow

so here exactly two sides will be yellow for only those small cubes which are on the edges of original cube minus those who are on the corner of those sides. Here there are 12 sides of the original cube.

And Number of such cubes = 12 * (9 - 2) = 84

so, probability of selecting a two sides painted yellow cube = 84/729 = 0.1152 or 11.52%