Question

Two dice are rolled together. The probability of getting an outcome \((x, y)\) where \(x>y\), is

• A. \(\frac{5}{12}\)
• B. \(\frac{5}{6}\)
• C. \(1\)
• D. \(0\)

Hint:

Instead of listing all outcomes, remember: for two dice, \(P(x>y) = P(x<y) = \frac{36 - 6}{2 \times 36} = \frac{15}{36}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When two dice are rolled, there are \(6 \times 6 = 36\) total possible outcomes. For any pair \((x, y)\), there are three possibilities: \(x = y\), \(x>y\), or \(x<y\).
Step 2: Detailed Explanation:
1. Case \(x = y\): The outcomes are \((1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\). Total = \(6\).
2. Total remaining outcomes where \(x \neq y\) is \(36 - 6 = 30\).
3. By symmetry, the number of outcomes where \(x>y\) must be equal to the number of outcomes where \(y>x\).
Number of outcomes where \(x>y = \frac{30}{2} = 15\).
4. Probability Calculation:
\[ P(x>y) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
\[ P(x>y) = \frac{15}{36} \]
Dividing by \(3\):
\[ P(x>y) = \frac{5}{12} \]
Step 3: Final Answer:
The probability is \(\frac{5}{12}\).