Find the probability of getting the sum of numbers on two dice (i) 9 (ii) at least 9, in one throw of a pair of dice.

Explanation

To solve the problem of finding the probability of getting the sum of numbers on two dice, we will break it down into two parts as specified in the question.

### Part (i): Finding the Probability of Getting a Sum of 9

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

2. **Identify Favorable Outcomes for a Sum of 9**:
We need to find all the combinations of the two dice that result in a sum of 9. The pairs that give a sum of 9 are:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

Thus, the favorable outcomes are: (3, 6), (4, 5), (5, 4), (6, 3). This gives us a total of 4 favorable outcomes.

3. **Calculate the Probability**:
The probability \( P \) of getting a sum of 9 is given by the formula:
\[ P(\text{sum} = 9) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{4}{36} = \frac{1}{9} \]

### Part (ii): Finding the Probability of Getting a Sum of at Least 9

1. **Identify Favorable Outcomes for a Sum of at Least 9**:
A sum of at least 9 includes sums of 9, 10, 11, and 12. We will list the combinations for each of these sums:
- **Sum = 9**: (3, 6), (4, 5), (5, 4), (6, 3) → 4 outcomes
- **Sum = 10**: (4, 6), (5, 5), (6, 4) → 3 outcomes
- **Sum = 11**: (5, 6), (6, 5) → 2 outcomes
- **Sum = 12**: (6, 6) → 1 outcome

Now, we sum all the favorable outcomes:
\[ \text{Total Favorable Outcomes} = 4 + 3 + 2 + 1 = 10 \]

2. **Calculate the Probability**:
The probability \( P \) of getting a sum of at least 9 is given by:
\[ P(\text{sum} \geq 9) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{10}{36} = \frac{5}{18} \]

### Final Answers:
- The probability of getting a sum of 9 is \( \frac{1}{9} \).
- The probability of getting a sum of at least 9 is \( \frac{5}{18} \).