three identical dice are rolled . Find the probability that the same number will appear on each of them .

Answer

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

### Step 1: Determine the total number of possible outcomes
When rolling three identical dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling three dice is calculated as follows:

\[ \text{Total outcomes} = 6 \times 6 \times 6 = 216 \]

### Step 2: Identify the favorable outcomes
Next, we need to find the number of favorable outcomes where all three dice show the same number. The possible outcomes where all dice show the same number are:

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

There are a total of 6 favorable outcomes.

### Step 3: Calculate the probability
The probability of an event is given by the formula:

\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Substituting the values we found:

\[ \text{Probability} = \frac{6}{216} \]

### Step 4: Simplify the probability
Now, we can simplify the fraction:

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

### Final Answer
Thus, the probability that the same number will appear on each of the three dice is:

\[ \frac{1}{36} \]

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