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: Use symmetry instead of listing every outcome for $x>y$.
Out of the 36 equally likely outcomes, exactly 6 have $x=y$ (the doubles).
Step 2: Split the remaining outcomes.
The remaining $36-6=30$ outcomes are split evenly between $x>y$ and $x<y$ by symmetry, since swapping the two dice values swaps one case into the other.
Step 3: Compute the favourable count.
\[ \text{Outcomes with } x>y = \frac{30}{2} = 15 \]
Step 4: Find the probability.
\[ P(x>y) = \frac{15}{36} = \frac{5}{12} \]
This matches option (1).
\[ \boxed{\frac{5}{12}} \]