A fair six-sided die is rolled twice. What is the probability that the sum of the two outcomes is equal to 9?

Introduction / Context:
Here we consider two rolls of a fair six-sided die and ask for the probability that the total of the two rolls equals 9. This is a standard discrete probability problem that reinforces the method of listing combinations that lead to a particular sum and then dividing by the total number of possible ordered outcomes.

Given Data / Assumptions:

• A fair die with faces numbered 1 to 6 is rolled twice.
• Each ordered pair of outcomes (first roll, second roll) is equally likely.
• Total number of ordered outcomes is 6 * 6 = 36.
• We require the probability that the sum of the two rolls is 9.

Concept / Approach:
The key idea is to list all pairs of integers (a, b) where a and b are between 1 and 6 and a + b = 9. Each of these is one favourable outcome. Then we divide the number of favourable outcomes by 36. Because the die is fair and the rolls are independent, this method is valid and straightforward.

Step-by-Step Solution:
All possible sums from two rolls range from 2 to 12. We are interested only in the sum 9. List all ordered pairs (a, b) such that a + b = 9 with 1 ≤ a, b ≤ 6. These pairs are: (3,6), (4,5), (5,4), (6,3). Number of favourable outcomes = 4. Total number of possible ordered outcomes = 36. Required probability = 4 / 36. Simplify 4 / 36 by dividing by 4 to get 1 / 9.

Verification / Alternative check:
We can check using the known distribution of sums for two dice. For sum 9, the known count is 4 outcomes, corresponding exactly to the pairs found: (3,6), (4,5), (5,4), (6,3). Dividing 4 by 36 gives 1/9. Since this matches both enumeration and the standard distribution table, the answer is confirmed.

Why Other Options Are Wrong:
1/6: This would correspond to 6 favourable outcomes, which is larger than the actual number 4.
1/2 and 3/4: These probabilities are far too large for one specific sum, as they would mean that half or three quarters of all outcomes yield a sum of 9.
None of these: This is incorrect because 1/9 is listed among the options and is correct.

Common Pitfalls:
A common mistake is to forget that the order of the two rolls matters, sometimes counting (3,6) and (6,3) as only one outcome. Another error is to try to compute the probability without listing pairs, leading to confusion about how many combinations actually give the sum 9. Systematic listing of pairs is the safest method in such questions.

Final Answer:
Thus, the probability that the sum of the two dice is 9 is 1/9.