Four identical dice are thrown. The probability that same number will appear on each of the die is

Options

A. `1/18`
B. `1/36`
C. `1/216`
D. `5/36`

Answer

To find the probability that the same number will appear on each of the four identical dice when they are thrown, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Outcomes:**
When throwing one die, there are 6 possible outcomes (1, 2, 3, 4, 5, or 6). Since we are throwing 4 dice, the total number of outcomes can be calculated as:
\[ \text{Total Outcomes} = 6 \times 6 \times 6 \times 6 = 6^4 \]
\[ 6^4 = 1296 \]

2. **Identify the Favorable Outcomes:**
We want to find the probability that all four dice show the same number. The favorable outcomes are:
- (1, 1, 1, 1)
- (2, 2, 2, 2)
- (3, 3, 3, 3)
- (4, 4, 4, 4)
- (5, 5, 5, 5)
- (6, 6, 6, 6)

There are a total of 6 favorable outcomes (one for each number).

3. **Calculate the Probability:**
The probability \( P \) of all four dice showing the same number is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{1296} \]

4. **Simplify the Probability:**
We can simplify the fraction:
\[ P = \frac{6}{1296} = \frac{1}{216} \]

### Final Answer:
The probability that the same number will appear on each of the four dice is:
\[ \frac{1}{216} \]