Let's solve this problem step by step.

Step 1: Total Possible Outcomes

When two dice are thrown simultaneously, each die has 6 faces, and each face can land in 6 possible ways. Therefore, the total number of outcomes when throwing two dice is:

$6 \times 6 = 36$

Step 2: Sum of Dice

The sum of the two dice can range from $2$ (if both dice show a 1) to $12$ (if both dice show a 6).

Step 3: Prime Numbers

We need to find all the prime numbers between 2 and 12. The prime numbers are:

$2, 3, 5, 7, 11$

Step 4: Outcomes for Prime Numbers

Now, we determine the number of outcomes for each of these prime sums:

• Sum = 2: The only combination is $(1,1)$ → 1 outcome.
• Sum = 3: The combinations are $(1,2), (2,1)$ → 2 outcomes.
• Sum = 5: The combinations are $(1,4), (2,3), (3,2), (4,1)$ → 4 outcomes.
• Sum = 7: The combinations are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ → 6 outcomes.
• Sum = 11: The combinations are $(5,6), (6,5)$ → 2 outcomes.

Total outcomes for prime numbers = $1 + 2 + 4 + 6 + 2 = 15$.

Step 5: Sums Greater Than 9

Next, we identify the sums greater than 9. These are:

$10, 11, 12$

We already know the outcomes for sum = 11 from the previous step.

• Sum = 10: The combinations are $(4,6), (5,5), (6,4)$ → 3 outcomes.
• Sum = 11: The combinations are $(5,6), (6,5)$ → 2 outcomes (already counted).
• Sum = 12: The only combination is $(6,6)$ → 1 outcome.

Total outcomes for sums greater than 9 = $3 + 2 + 1 = 6$.

Step 6: Combined Outcomes

We now combine the two sets of outcomes:

• Outcomes for prime sums: 15 outcomes
• Outcomes for sums greater than 9: 6 outcomes

However, the outcome for sum = 11 has been counted in both sets, so we subtract that overlap. The number of overlapping outcomes is 2 (the pair $(5,6)$ and $(6,5)$).

So, the total number of favorable outcomes is:

$15 + 6 - 2 = 19$

Step 7: Probability

The probability is the number of favorable outcomes divided by the total number of possible outcomes:

$\frac{19}{36}$

Final Answer:

The probability is $\frac{19}{36}$, which doesn't match any of the provided options exactly.

However, the most relevant part of the question seems to be about interpreting the provided options. Based on your provided answers, none of the answers seem to directly match the correct probability.

Would you like to discuss this further or clarify the options?