Two dice are thrown simultaneously. Find the probability that :
the total of the numbers on the dice is greater than 9.

Answer

To find the probability that the total of the numbers on two thrown dice is greater than 9, we can follow these steps:

### Step 1: Determine the Total Outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify Favorable Outcomes
Next, we need to find the combinations of the numbers on the dice that result in a sum greater than 9. The possible sums greater than 9 are 10, 11, and 12. We can list the combinations for each of these sums:

- **Sum = 10:**
- (4, 6)
- (5, 5)
- (6, 4)

- **Sum = 11:**
- (5, 6)
- (6, 5)

- **Sum = 12:**
- (6, 6)

Now, let's count the favorable outcomes:
- For sum = 10, we have 3 combinations.
- For sum = 11, we have 2 combinations.
- For sum = 12, we have 1 combination.

Adding these gives us:
\[ \text{Favorable Outcomes} = 3 + 2 + 1 = 6 \]

### Step 3: Calculate the Probability
The probability \( P \) of an event is given by the formula:
\[ P(\text{Event}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} \]
Substituting the values we found:
\[ P(\text{Sum > 9}) = \frac{6}{36} = \frac{1}{6} \]

### Final Answer
The probability that the total of the numbers on the dice is greater than 9 is:
\[ \frac{1}{6} \]

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