Question

A 5-by-5-by-5 wooden cube is cut into 125 smaller cubes after all six sides being painted red. If a smaller cube is chosen at random, then rolled like a die, what is the probability that a red face will be on top? Give your answer as a common faction.

Answer 1

We are giving a cube with dimentions

5x5x5

to cut into 125 equals cubes we have to make 5 horizontal cuts and 5 vertical cuts

now of these 125 cubes we have

8 are corner cubes, then they are painted red on three sides. ( 3 out of 6 sides)

3*3*3= 27 are interior cubes then they have no sides painted red (0 out of 6 sides)

9*6 =54 are cubes in front painted only one time ( 1 out of 6 sides)

12*3=36 are cubes painted 2 times(2 out of 6 sides)

then The probability of a red side facing up is giving by

$P=\left(8/125\right)\left(3/6\right)+\left(27/125\right)\left(0/6\right)+\left(54/125\right)\left(1/6\right)+\left(36/125\right)\left(2/6\right)$$P=\left(8/125\right)\left(3/6\right)+\left(27/125\right)\left(0/6\right)+\left(54/125\right)\left(1/6\right)+\left(36/125\right)\left(2/6\right)$

then the probability of a red side facing up is

$P=\frac{1}{5}$

P=0.2 or 20%