### Instruction: Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction.### Response: I need to find the number of favorable outcomes and divide it by the total number of possible outcomes.
The total number of possible outcomes is $6 \times 6 = 36$, since each roll has $6$ choices and they are independent.
The favorable outcomes are the ones where the first roll and the second roll differ by $2$.
I can list them as pairs: $(1, 3), (3, 1), (2, 4), (4, 2), (3, 5), (5, 3), (4, 6), (6, 4)$.
There are $8$ pairs that satisfy the condition, so the probability is $\frac{8}{36}$.
I can simplify this fraction by dividing both the numerator and the denominator by $4$, which gives me $\frac{2}{9}$.