Question

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

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

Hint:

By symmetry, the number of outcomes where $x \gt y$ is equal to the number of outcomes where $y \gt x$.
The total number of outcomes is 36. The number of outcomes where $x = y$ (doubles) is 6.
The remaining outcomes are $36 - 6 = 30$. Dividing by 2 gives exactly 15 outcomes where $x \gt y$. This is a great shortcut!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Two fair six-sided dice are rolled simultaneously.
We need to find the probability that the outcome $(x, y)$ on the two dice satisfies the inequality $x \gt y$, where $x$ is the value on the first die and $y$ is the value on the second die.

Step 2: Key Formula or Approach:
The probability of an event $E$ is given by:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
The total number of possible outcomes when two dice are rolled is $6 \times 6 = 36$.
We will count all outcomes where the number on the first die is strictly greater than the number on the second die.

Step 3: Detailed Explanation:

• Write down the total sample space size:
\[ n(S) = 6 \times 6 = 36 \]

• List all possible outcomes $(x, y)$ and identify those where $x \gt y$:
- For $x = 1$: No possible values of $y$ since $y \ge 1$.
- For $x = 2$: $(2, 1)$ (1 outcome)
- For $x = 3$: $(3, 1), (3, 2)$ (2 outcomes)
- For $x = 4$: $(4, 1), (4, 2), (4, 3)$ (3 outcomes)
- For $x = 5$: $(5, 1), (5, 2), (5, 3), (5, 4)$ (4 outcomes)
- For $x = 6$: $(6, 1), (6, 2), (6, 3), (6, 4), (6, 5)$ (5 outcomes)

• Count the total number of favorable outcomes $n(E)$:
\[ n(E) = 1 + 2 + 3 + 4 + 5 = 15 \]

• Compute the probability:
\[ P(E) = \frac{15}{36} = \frac{5}{12} \]


Step 4: Final Answer:
The probability of getting an outcome $(x, y)$ where $x \gt y$ is $\frac{5}{12}$.