Let's first list all the possible sums when rolling two different dice. Each die has faces showing the numbers: 1, 2, 3, 5, 7, and 9.

Step 1: Find all possible sums

We need to calculate the sum of every pair of faces from the two dice. Since the dice are distinct, we consider every combination of one number from the first die and one from the second die.

The possible sums are:

• 1 + 1 = 2
• 1 + 2 = 3
• 1 + 3 = 4
• 1 + 5 = 6
• 1 + 7 = 8
• 1 + 9 = 10
• 2 + 2 = 4
• 2 + 3 = 5
• 2 + 5 = 7
• 2 + 7 = 9
• 2 + 9 = 11
• 3 + 3 = 6
• 3 + 5 = 8
• 3 + 7 = 10
• 3 + 9 = 12
• 5 + 5 = 10
• 5 + 7 = 12
• 5 + 9 = 14
• 7 + 7 = 14
• 7 + 9 = 16
• 9 + 9 = 18

Step 2: Identify the distinct sums

From this list, the distinct sums are:

$2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18$

So, there are 14 different sums.

Step 3: Find the most likely sum

The most likely sum is the one that appears the most often. From the list above:

• The sum 10 appears 3 times (1 + 9, 3 + 7, 5 + 5).
• The sum 12 appears 2 times (3 + 9, 5 + 7).
• Other sums appear less frequently.

Thus, the most likely sum is 10.

Final Answer:

• 14 different sums should occur.
• The most likely sum is 10.

Would you like further details or clarifications on the calculation? Here are 5 related questions to explore more:

1. How would the result change if we used dice with different sets of numbers, e.g., 1, 2, 3, 4, 5, 6?
2. How does the likelihood of a sum relate to how frequently it appears in all combinations?
3. What would happen if you rolled three dice instead of two?
4. How can you calculate the total number of possible outcomes in a dice roll?
5. What role does symmetry play in calculating the most likely sums in dice problems?

Tip: To understand probability distributions better, remember that sums with more combinations are typically more likely, as they can be reached in multiple ways.