A cube painted red on all sides, is cut into `125` equal small cubes. A small cube when picked up is found to show red colour on one of its faces. Find the probability that two more faces also show red colour.

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript while providing a clear and structured explanation.

### Step 1: Understand the Problem
We have a cube that is painted red on all sides and then cut into 125 smaller cubes. We need to find the probability that if a small cube shows red on one face, it also shows red on two more faces.

### Step 2: Determine the Size of the Cube
Since the cube is cut into 125 smaller cubes, we can find the dimensions of the original cube. The cube root of 125 is 5, meaning the original cube is a \(5 \times 5 \times 5\) cube.

### Step 3: Calculate the Total Number of Small Cubes
The total number of small cubes is given as 125.

### Step 4: Identify the Types of Small Cubes
1. **Corner Cubes**: These cubes have 3 faces painted red. There are 8 corners in a cube.
2. **Edge Cubes (not corners)**: These cubes have 2 faces painted red. Each edge of the cube has \(5 - 2 = 3\) small cubes (excluding the corners), and since there are 12 edges, the total number of edge cubes is \(12 \times 3 = 36\).
3. **Face Cubes (not on edges)**: These cubes have 1 face painted red. Each face of the cube has \(3 \times 3 = 9\) small cubes (excluding the edges), and since there are 6 faces, the total number of face cubes is \(6 \times 9 = 54\).
4. **Inner Cubes**: These cubes have no red paint. The inner cube is \(3 \times 3 \times 3\), which gives us \(27\) inner cubes.

### Step 5: Calculate the Total Number of Painted Cubes
The total number of cubes that have at least one face painted red is:
\[ 125 - 27 = 98 \]

### Step 6: Find the Favorable Outcomes
We are interested in the small cubes that show red on one face and also have red on two more faces. These are the corner cubes, as they have 3 faces painted red. The number of corner cubes is \(8\).

### Step 7: Calculate the Probability
The probability \(P\) that a randomly selected small cube, which shows red on one face, also shows red on two more faces is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of corner cubes}}{\text{Total number of painted cubes}} = \frac{8}{98} \]
This can be simplified:
\[ P = \frac{4}{49} \]

### Final Answer
The probability that two more faces also show red color is \(\frac{4}{49}\).

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