Malia puts together six 4-by-4-by-1 inch square layers with 1-inch cubes, then stacks
them up to form a rectangular prism. This prism rests on a wooden table, on its square
base. All the 1-inch cubes Malia used are white except one, which is red. As a reduced
fraction, what is the probability that the red cube is visible if it is located at random?
To find the probability that the red cube is visible, we need to determine the total number of cubes in the rectangular prism, as well as the number of cubes that can potentially hide the red cube.
First, let's calculate the total number of cubes in the rectangular prism.
Given that there are 6 layers, and each layer is a 4-by-4 grid of 1-inch cubes, we can calculate the number of cubes in each layer as 4 * 4 = 16.
Since there are 6 layers, the total number of cubes is 6 * 16 = 96.
Next, let's determine the number of cubes that can potentially hide the red cube.
Since the rectangular prism rests on its square base, there will be 16 cubes on the top layer.
Out of these 16 cubes, only one is red.
Therefore, there are 16 - 1 = 15 cubes that can hide the red cube.
Now, to calculate the probability, we divide the number of cubes that can hide the red cube by the total number of cubes.
The probability is given by: probability = (number of favorable outcomes) / (total number of outcomes)
In this case, the favorable outcome is the red cube being visible, so the number of favorable outcomes is 1.
The total number of outcomes is 96.
Therefore, the probability that the red cube is visible is 1/96.
As a reduced fraction, this is the final answer.