Two fair six-sided dice are rolled one time. What is the probability that the sum of the numbers showing on...

1. INFER the total number of possible outcomes

• Given information:
  • Two fair six-sided dice
  • Each die shows: \(\{1, 2, 3, 4, 5, 6\}\)
• Since the dice rolls are independent, total outcomes = \(6 \times 6 = 36\)

2. CONSIDER ALL CASES to find favorable outcomes

• We need sum = 2, so \(\mathrm{die}_1 + \mathrm{die}_2 = 2\)
• Let's systematically check:
  • If \(\mathrm{die}_1 = 1\), then \(\mathrm{die}_2\) must equal 1 (since \(1 + 1 = 2\)) ✓
  • If \(\mathrm{die}_1 = 2\), then \(\mathrm{die}_2\) must equal 0 (impossible - dice don't have 0)
  • If \(\mathrm{die}_1 \geq 3\), then \(\mathrm{die}_2\) would need to be negative (impossible)
• Only one favorable outcome: \((1,1)\)

3. INFER the final probability

• Apply the probability formula:

\(\mathrm{P(sum = 2)} = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{1}{36}\)

Answer: A (1/36)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students don't systematically think through what combinations give sum = 2. They might hastily assume there are multiple ways to get this sum, similar to higher sums like 7 or 8, without carefully checking what's actually possible with the constraint that each die shows 1-6.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students correctly identify there's one way to get sum = 2, but incorrectly calculate total possible outcomes. They might think each die contributes 6 outcomes for a total of \(6 + 6 = 12\), rather than using the multiplication principle to get \(6 \times 6 = 36\).

This may lead them to select Choice B (1/12).

The Bottom Line:

This problem tests whether students can systematically analyze all possible outcomes in a simple probability scenario. The key insight is recognizing that sum = 2 is the smallest possible sum with two standard dice, making it have very few (in this case, exactly one) ways to occur.