Two fair dice are rolled simultaneously. What is the probability that the sum of the numbers appearing on the two dice is exactly 9?

Introduction / Context:
Here you must work with the sample space of two dice and identify which ordered pairs give a specific total, in this case a sum of 9. Problems involving sums of dice outcomes are very common in probability sections and help you practise systematic counting of favorable outcomes.

Given Data / Assumptions:

• Two fair six sided dice are rolled together.
• Each die shows one of 1, 2, 3, 4, 5, 6.
• All 36 ordered pairs of outcomes are equally likely.
• We need the probability that the sum of the two results is 9.

Concept / Approach:
We must identify all ordered pairs (a, b) with a from 1 to 6 and b from 1 to 6 such that a + b = 9. Once we have counted all such pairs, the probability will be: Probability = (Number of pairs with sum 9) / 36. This direct counting strategy is straightforward and avoids confusion about symmetry.

Step-by-Step Solution:
Step 1: List all pairs of numbers between 1 and 6 that add to 9. Step 2: The valid pairs are: (3, 6), (4, 5), (5, 4), and (6, 3). Step 3: Count these pairs. There are 4 such ordered outcomes. Step 4: Total possible ordered outcomes when rolling two dice = 6 * 6 = 36. Step 5: Probability that the sum is 9 = number of favorable outcomes / total outcomes = 4 / 36. Step 6: Simplify 4 / 36 by dividing numerator and denominator by 4 to get 1 / 9.

Verification / Alternative check:
You can also reason by symmetry. The smallest possible sum of two dice is 2 and the largest is 12. For sums in the middle, such as 7, 8, and 9, there are several combinations, but as the target sum moves away from 7 the number of combinations decreases. Specifically, 7 has the most combinations, and 8 and 9 have equal numbers of combinations to each other. By listing systematically, you confirm that there are 4 combinations for 9, so the probability 4/36 = 1/9 is consistent with this pattern.

Why Other Options Are Wrong:
Option 1/6 would require 6 favorable outcomes, which is not true here. Option 2/9 and 4/9 are larger than the correct probability and would imply many more pairs summing to 9 than actually exist. Option 5/18 is also too large and does not correspond to any correct counting pattern. Only 1/9 matches the actual ratio of favorable to total outcomes based on the list of valid pairs.

Common Pitfalls:
Students sometimes forget that the order of the dice matters when counting outcomes, so they may count (3,6) and (6,3) as a single outcome instead of two. Others may mistakenly include pairs that sum to 8 or 10 when working quickly. To avoid such mistakes, write down the pairs explicitly and check each sum. Remember that 36 is the total number of ordered outcomes for two fair six sided dice, and keep this constant in your denominator for such problems.

Final Answer:
The probability that the sum of the numbers on the two dice is 9 is 1/9.