The correct sample space $S$ for the random experiment where a pair of dice is rolled once and the sum of the dots on the two faces is recorded would include all possible sums that can occur when rolling two dice.

Here’s the reasoning:

1. The minimum sum possible is $2$ (if both dice show 1).
2. The maximum sum possible is $12$ (if both dice show 6).

The possible sums when rolling two dice are: $2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12$

Conclusion:

The correct answer is: $S = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$

This option includes all possible sums without omitting or repeating any.

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Would you like more details on this? Here are some related questions you may find interesting:

1. How many total outcomes are there when rolling two dice?
2. What is the probability of rolling a sum of 7 with two dice?
3. How can the probability distribution of the sums of two dice be represented?
4. What is the probability of rolling an even sum with two dice?
5. How many ways can a sum of 10 be rolled with two dice?

Tip: When determining sample spaces for dice rolls, always consider the range of outcomes for each die and use combinations to ensure all possible results are included.