A special dice with numbers 1,-1, 2, -2.0 and 3 is thrown thrice. What is the probability that the sum of the numbers occurring on the upper face is zero?

Options

A. `(1)/(72)`
B. `(1)/(8)`
C. `(7)/(12)`
D. `(25)/(216)`

Answer

To solve the problem of finding the probability that the sum of the numbers occurring on the upper face of a special die (with faces numbered 1, -1, 2, -2, 0, and 3) thrown three times equals zero, we will follow these steps:

### Step 1: Determine the Total Outcomes
When a die is thrown, there are 6 possible outcomes for each throw. Since the die is thrown three times, the total number of outcomes can be calculated as:
\[ \text{Total Outcomes} = 6^3 = 216 \]

### Step 2: Identify Favorable Outcomes
We need to find all the combinations of numbers that can sum to zero. The possible numbers on the die are 1, -1, 2, -2, 0, and 3. We will look for combinations of these numbers that yield a sum of zero.

1. **Combination of (1, -1, 0)**:
- Possible combinations: (1, -1, 0), (-1, 1, 0), (0, 1, -1), (0, -1, 1), etc.
- The arrangements can be calculated as \(3!\) (since all three numbers are different).
- Total combinations = 6.

2. **Combination of (2, -2, 0)**:
- Similar to the previous case, we can arrange these three numbers in \(3!\) ways.
- Total combinations = 6.

3. **Combination of (1, -1, -2)**:
- The arrangements can be calculated as \(3!\).
- Total combinations = 6.

4. **Combination of (-1, -1, 2)**:
- Here, we have two identical numbers (-1).
- The arrangements can be calculated as \(\frac{3!}{2!} = 3\).
- Total combinations = 3.

5. **Combination of (1, -2, -1)**:
- This is similar to the previous case, and we can arrange these three numbers in \(3!\) ways.
- Total combinations = 6.

6. **Combination of (0, 0, 0)**:
- There is only one way to arrange three zeros.
- Total combinations = 1.

### Step 3: Calculate Total Favorable Outcomes
Now, we will sum all the favorable outcomes:
\[ \text{Total Favorable Outcomes} = 6 + 6 + 6 + 3 + 6 + 1 = 28 \]

### Step 4: Calculate the Probability
The probability \(P\) that the sum of the numbers is zero can be calculated using the formula:
\[ P = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} = \frac{28}{216} \]
This fraction can be simplified:
\[ P = \frac{7}{54} \]

### Conclusion
Thus, the probability that the sum of the numbers occurring on the upper face of the die is zero is:
\[ \frac{7}{54} \]