When two fair dice are thrown simultaneously, what is the probability that the sum of the two numbers obtained is less than 12?

Introduction / Context:
This question deals with the sample space of two fair dice. Each die has 6 faces and all ordered pairs of outcomes are equally likely. We are asked for the probability that the sum of the two numbers is less than 12. The maximum possible sum when two dice are thrown is 12, which occurs only when both dice show 6. Recognising this allows us to solve the problem very quickly using a complement argument.

Given Data / Assumptions:

Two fair six faced dice are thrown at the same time.

Each die can show an integer from 1 to 6.

All 36 ordered pairs (1,1) to (6,6) are equally likely.

We are interested in the event that the sum of the two numbers is less than 12.

Concept / Approach:
Instead of counting all the sums less than 12 directly, it is easier to compute the probability of the complement event and subtract from 1. The only way the sum can be 12 is if both dice show 6. Thus, P(sum = 12) is straightforward to compute. Then P(sum less than 12) = 1 - P(sum = 12). This is a standard example of using the complement rule in probability to simplify counting.

Step-by-Step Solution:
Step 1: Total number of possible ordered outcomes when two dice are thrown = 6 * 6 = 36.Step 2: The maximum sum is 12, which occurs only when the first die is 6 and the second die is 6.Step 3: There is exactly 1 ordered outcome that gives a sum of 12, namely (6, 6).Step 4: Probability that the sum is 12 = 1 / 36.Step 5: The event that the sum is less than 12 is the complement of the event that the sum is 12, because every other outcome has sum from 2 to 11.Step 6: Therefore, P(sum less than 12) = 1 - P(sum = 12) = 1 - 1 / 36.Step 7: Compute 1 - 1 / 36 = 35 / 36.

Verification / Alternative check:
If desired, we could check explicitly that all sums from 2 to 11 are achievable and that at least one outcome leads to each of these sums. For any given sum from 2 to 11, you can find at least one ordered pair (such as (1,1) for 2, (1,2) for 3, and so on). Since 36 total outcomes exist and exactly one outcome gives sum 12, the other 35 outcomes must correspond to sums less than 12, validating the complement method.

Why Other Options Are Wrong:
The fractions 17/36 and 15/36 correspond to much smaller subsets of outcomes and would imply that many sums greater than or equal to some value have been excluded incorrectly. The value 1/36 is actually the probability of getting a sum of exactly 12, not less than 12. The fraction 5/36 is also much too small and would mean that only a few outcomes satisfy the condition, which is not true here.

Common Pitfalls:
Some learners mistakenly count only a subset of sums or miscount the total sample space, forgetting that the dice outcomes are ordered pairs. Others confuse the event "sum less than 12" with "sum less than or equal to some smaller number" and undercount favourable outcomes. Using the complement rule is a powerful and safe approach whenever the complement event has a very simple structure, as in this case.

Final Answer:
The probability that the sum of the two dice is less than 12 is 35/36.