Three fair cubical dice are thrown simultaneously. The probability that all three dice have the same number of dots on the faces showing up is:

Question

Accepted Answer

To determine the probability that all three dice show the same number of dots, we need to consider the total number of possible outcomes and the number of favorable outcomes.

1. **Total Number of Outcomes:**
   - Each die has 6 faces, and since three dice are thrown, the total number of possible outcomes is $6 \times 6 \times 6 = 216$.

2. **Favorable Outcomes:**
   - For all three dice to show the same number, they must all show one of the numbers from 1 to 6. There are 6 possible outcomes where all three dice show the same number: (1,1,1), (2,2,2), (3,3,3), (4,4,4), (5,5,5), and (6,6,6). Thus, there are 6 favorable outcomes.

3. **Probability Calculation:**
   - The probability of all three dice showing the same number is the number of favorable outcomes divided by the total number of outcomes. Therefore, the probability is $\frac{6}{216} = \frac{1}{36}$.

Thus, the correct answer is Option 3: $\frac{1}{36}$.

**Explanation for Incorrect Options:**

- **Option 1: $\frac{1}{108}$**
  - This option suggests that there are fewer favorable outcomes or more total outcomes than there actually are. Since the calculation shows $\frac{1}{36}$, $\frac{1}{108}$ is incorrect because it underestimates the probability.

- **Option 2: $\frac{1}{72}$**
  - This option also underestimates the probability. It implies a different number of favorable outcomes or total outcomes, which does not match the correct calculation of $\frac{1}{36}$.

- **Option 4: $\frac{1}{54}$**
  - This option overestimates the probability. It suggests a higher probability than calculated, which is incorrect based on the actual number of favorable and total outcomes.

In conclusion, the correct probability that all three dice show the same number is $\frac{1}{36}$, which is why Option 3 is the correct answer.