To solve this problem, we need to find the probability that the sum of the two dice is either 3 or 9.

Step 1: Calculate the Total Possible Outcomes

Since each die has 6 faces, rolling two dice gives a total of: $6 \times 6 = 36 \text{ possible outcomes}$ This matches the grid shown in the image, with 36 pairs listed.

Step 2: Find Outcomes Where the Sum is 3

Possible pairs that sum to 3 are:

1. (1, 2)
2. (2, 1)

There are 2 outcomes where the sum is 3.

Step 3: Find Outcomes Where the Sum is 9

Possible pairs that sum to 9 are:

1. (3, 6)
2. (4, 5)
3. (5, 4)
4. (6, 3)

There are 4 outcomes where the sum is 9.

Step 4: Combine Outcomes

The outcomes with a sum of either 3 or 9 are: $2 + 4 = 6 \text{ outcomes}$

Step 5: Calculate the Probability

The probability of getting a sum of 3 or 9 is given by: $\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6}$

Answer

The probability that the sum of the dice is 3 or 9 is: $\frac{1}{6}$

Would you like more details on how these sums were identified or have other questions?

Here are five related questions to expand your understanding:

1. What is the probability of rolling a sum of 7 with two dice?
2. How would the probability change if we wanted a sum of 4 or 8?
3. If we rolled three dice, what would the probability be for a sum of 9?
4. How many outcomes give a sum greater than 8 with two dice?
5. What is the probability of rolling doubles (like (1,1), (2,2), etc.)?

Tip: When calculating probabilities with dice, listing all possible outcomes helps ensure no combinations are missed.