Two fair six faced dice are thrown once. What is the probability that the sum of the numbers appearing on the upper faces is exactly 9?

Introduction / Context:
This problem is about the distribution of sums when two standard dice are rolled. Each die has faces numbered from 1 to 6, and we want the probability that the sum of the two numbers equals 9. Such questions help learners become familiar with listing pairs of outcomes and understanding how some sums are more likely than others.

Given Data / Assumptions:

• Two fair six sided dice are rolled simultaneously.
• Each die can show an integer from 1 to 6.
• All 36 ordered pairs of outcomes are equally likely.
• We seek the probability that the sum of the two results equals 9.

Concept / Approach:
The key idea is to count the number of outcome pairs that give the desired sum. Each outcome is an ordered pair (first die, second die). We list all pairs whose components add to 9 and then divide by the total number of possible outcomes, which is 36. This direct counting method is efficient for sums in dice problems.

Step-by-Step Solution:
Step 1: Total possible ordered outcomes when two dice are rolled is 6 * 6 = 36.Step 2: List all pairs of numbers (x, y) with x from 1 to 6 and y from 1 to 6 such that x + y = 9.Step 3: The valid pairs are (3, 6), (4, 5), (5, 4) and (6, 3).Step 4: Count these pairs and observe that there are 4 favorable outcomes.Step 5: Probability of obtaining a sum of 9 is therefore 4 / 36.Step 6: Simplify 4 / 36 by dividing numerator and denominator by 4 to get 1 / 9.
Verification / Alternative check:
We can verify by noting that the most likely sums are around 7, and that sums like 2 and 12 have only one combination each. For sum 9, we reasonably expect a moderate number of combinations, and our list of 4 pairs is consistent with this pattern. Another check is to ensure that we have not missed any pair such as (2, 7) or (7, 2), but these are impossible because each die only goes up to 6. Thus the enumeration appears complete and correct.

Why Other Options Are Wrong:
1/2 and 3/4 are much too large and do not reflect the limited number of favorable outcomes compared to 36 total possibilities. 2/9 would correspond to 8 favorable outcomes, which is double our correctly counted number. 5/36 would require exactly 5 favorable pairs, but we have found only 4 valid pairs that add up to 9.

Common Pitfalls:
Students sometimes miscount by forgetting the ordered nature of outcomes, or by including impossible combinations such as 7 on a single die. Some may also forget that (4, 5) and (5, 4) are distinct outcomes when the dice are distinguishable. Being systematic in listing all possible pairs and checking the sum helps avoid such mistakes.

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