What is the probability that the sum of two dice rolls will yield a 5, and then when both are thrown again, their sum will yield a 9?

Answer

To solve the problem of finding the probability that the sum of two dice rolls will yield a 5, and then when both are thrown again, their sum will yield a 9, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Outcomes for Two Dice:**
- Each die has 6 faces. Therefore, when rolling two dice, the total number of possible outcomes is:
\[ 6 \times 6 = 36 \]

2. **Find the Number of Ways to Get a Sum of 5:**
- The pairs of dice rolls that yield a sum of 5 are:
- (1, 4)
- (2, 3)
- (3, 2)
- (4, 1)
- Thus, there are 4 favorable outcomes for getting a sum of 5.

3. **Calculate the Probability of Getting a Sum of 5:**
- The probability \( P(A) \) of getting a sum of 5 is given by the formula:
\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{36} = \frac{1}{9} \]

4. **Find the Number of Ways to Get a Sum of 9:**
- The pairs of dice rolls that yield a sum of 9 are:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)
- Thus, there are also 4 favorable outcomes for getting a sum of 9.

5. **Calculate the Probability of Getting a Sum of 9:**
- The probability \( P(B) \) of getting a sum of 9 is given by:
\[ P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{36} = \frac{1}{9} \]

6. **Combine the Probabilities of Independent Events:**
- Since the two events (getting a sum of 5 and then getting a sum of 9) are independent, we multiply their probabilities:
\[ P(A \text{ and } B) = P(A) \times P(B) = \frac{1}{9} \times \frac{1}{9} = \frac{1}{81} \]

### Final Answer:
The probability that the sum of two dice rolls will yield a 5, and then when both are thrown again, their sum will yield a 9 is:
\[ \frac{1}{81} \]