Suppose we roll a six-sided die twice, each time recording the value that lands face up. Let X be the sum of the two rolls. What is the support of X?

Question

Suppose we roll a six-sided die twice, each time recording the value that lands face up.  Let X be the sum of the two rolls.  What is the support of X?

BlackTom AI · Accepted Answer

Two rolls produce ordered pairs (i, j) with i, j ∈ {1,2,3,4,5,6}, and X = i + j.

1) Relevant knowledge points and concepts
- Support of a discrete random variable: the set of values that the variable can take with positive probability.
- Each die roll can be 1 through 6, so the sum X must lie between the minimum possible sum and the maximum possible sum.

2) Complete reasoning process, step by step
- Minimum sum: smallest possible outcomes on both rolls are 1 and 1, so min X = 1 + 1 = 2.
- Maximum sum: largest possible outcomes on both rolls are 6 and 6, so max X = 6 + 6 = 12.
- We must show every integer between 2 and 12 is attainable (has positive probability).

3) Intermediate steps / constructive existence for each sum
- For any s with 2 ≤ s ≤ 7, choose (1, s − 1). Since s − 1 ∈ {1,...,6} in this range, that ordered pair is valid, so X = s occurs.
- For any s with 8 ≤ s ≤ 12, choose (6, s − 6). Since s − 6 ∈ {2,...,6} in this range, that ordered pair is valid, so X = s occurs.
- Examples: 2 = (1,1); 3 = (1,2) or (2,1); 7 = (1,6) or (6,1); 12 = (6,6).

4) Why the correct answer is correct
- Because the smallest possible sum is 2 and largest is 12, and we have shown explicit ordered pairs giving every integer between them, the support is exactly {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

5) Why other options are incorrect
- "1/6", "1/12", "1/36" are single probability values, not sets of possible sums (so not supports).
- "{1,2,3,4,5,6}" lists single-roll outcomes, not sums of two rolls; sums cannot be 1.
- "{2,3,4,...,36}" extends beyond the possible maximum sum (36 would be product or misinterpretation); the sum cannot exceed 12.

6) Conclusion / final confirmation
- The support of X is {2, 3, 4, ..., 12}, which matches the provided correct answer.

Suppose we roll a six-sided die twice, each time recording the value that lands face up. Let X be the sum of the two rolls. What is the support of X?单项选择题

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