When three identical dice are rolled, then what is the probability of getting same number on each dice?

Options

A. `(1)/(6)`
B. `(1)/(2)`
C. `(1)/(216)`
D. `(1)/(36)`

Answer

To find the probability of getting the same number on each of the three identical dice when rolled, we can follow these steps:

### Step 1: Determine the Total Outcomes
When rolling a single die, there are 6 possible outcomes (1, 2, 3, 4, 5, or 6). Since we are rolling three identical dice, the total number of outcomes can be calculated as follows:

\[ \text{Total Outcomes} = 6 \times 6 \times 6 = 6^3 = 216 \]

### Step 2: Determine the Favorable Outcomes
Favorable outcomes are those where all three dice show the same number. The possible combinations where all three dice show the same number are:

- (1, 1, 1)
- (2, 2, 2)
- (3, 3, 3)
- (4, 4, 4)
- (5, 5, 5)
- (6, 6, 6)

Thus, there are a total of 6 favorable outcomes.

### Step 3: Calculate the Probability
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, we can calculate the probability as follows:

\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{216} \]

### Step 4: Simplify the Probability
Now, we simplify the fraction:

\[ \frac{6}{216} = \frac{1}{36} \]

### Conclusion
The probability of getting the same number on each of the three dice when rolled is:

\[ \text{Required Probability} = \frac{1}{36} \]

---