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

To solve the problem of finding the probability that the outcome \((x, y)\) from rolling two dice satisfies \(x > y\), we follow these steps:

1. Determine the total number of possible outcomes when two dice are rolled. Each die has 6 faces, leading to a total of \(6 \times 6 = 36\) outcomes. 
2. Find the number of favorable outcomes for the condition \(x > y\).
   • If \(x = 2\), possible \(y\) values are \(1\). So, 1 favorable outcome: \((2, 1)\).
   • If \(x = 3\), possible \(y\) values are \(1, 2\). So, 2 favorable outcomes: \((3, 1)\), \((3, 2)\).
   • If \(x = 4\), possible \(y\) values are \(1, 2, 3\). So, 3 favorable outcomes: \((4, 1)\), \((4, 2)\), \((4, 3)\).
   • If \(x = 5\), possible \(y\) values are \(1, 2, 3, 4\). So, 4 favorable outcomes: \((5, 1)\), \((5, 2)\), \((5, 3)\), \((5, 4)\).
   • If \(x = 6\), possible \(y\) values are \(1, 2, 3, 4, 5\). So, 5 favorable outcomes: \((6, 1)\), \((6, 2)\), \((6, 3)\), \((6, 4)\), \((6, 5)\).
3. Calculate the probability using the formula: 

\[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{15}{36} = \frac{5}{12}.\]

Therefore, the probability of getting \((x, y)\) where \(x > y\) is \(\frac{5}{12}\).

The correct answer is: \(\frac{5}{12}\).