Two dice are rolled simultaneously. Find the probability of :
obtaining a total of at least 9.

Answer

To find the probability of obtaining a total of at least 9 when two dice are rolled, we can follow these steps:

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

### Step 2: Identify Favorable Outcomes
Next, we need to find the combinations of the two dice that result in a total of at least 9. The possible sums and their combinations are as follows:

- **Sum = 9**:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

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

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

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

Now, let's count the total favorable outcomes:
- For a sum of 9: 4 outcomes
- For a sum of 10: 3 outcomes
- For a sum of 11: 2 outcomes
- For a sum of 12: 1 outcome

Adding these gives us:
\[ \text{Total favorable outcomes} = 4 + 3 + 2 + 1 = 10 \]

### Step 3: Calculate the Probability
The probability \( P \) of obtaining a total of at least 9 is given by the formula:
\[ P(\text{at least 9}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{36} \]

### Step 4: Simplify the Probability
We can simplify the fraction:
\[ P(\text{at least 9}) = \frac{10}{36} = \frac{5}{18} \]

### Final Answer
Thus, the probability of obtaining a total of at least 9 when two dice are rolled is:
\[ \frac{5}{18} \]

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