Ruby and Sam each roll a fair 6 -sided die with the numbers $1,2,3,4,5$, and 6 on its faces. Sam subtracts the number on his roll from the number on Ruby's roll. What is the probability that the result is a negative number?
(A) $\frac{5}{18}$
(B) $\frac{5}{12}$
(C) $\frac{7}{12}$
(D) $\frac{1}{2}$
(E) $\frac{5}{6}$

## Part C: Each correct answer is worth 8.
Solution 1

The table below shows the possible differences between the number on Ruby's roll and the number on Sam's roll.

|  |  | Number on Ruby's Roll |  |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| ![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-058.jpg?height=571&width=61&top_left_y=1168&top_left_x=293) |  | 1 | 2 | 3 | 4 | 5 | 6 |
|  | 1 | 0 | 1 | 2 | 3 | 4 | 5 |
|  | 2 | -1 | 0 | 1 | 2 | 3 | 4 |
|  | 3 | -2 | -1 | 0 | 1 | 2 | 3 |
|  | 4 | -3 | -2 | -1 | 0 | 1 | 2 |
|  | 5 | -4 | -3 | -2 | -1 | 0 | 1 |
|  | 6 | -5 | -4 | -3 | -2 | -1 | 0 |

Looking at the table above, the total number of possible outcomes is $6 \times 6=36$.

The total number of outcomes for which the difference is negative is 15 .

The probability that the result from subtracting the number on Sam's roll from the number on Ruby's roll is negative is $\frac{15}{36}=\frac{5}{12}$.

## Solution 2

Ruby and Sam each have 6 possible outcomes when they roll the dice, and so the total number of possible outcomes is $6 \times 6=36$.

Of these 36 possible outcomes, there are 6 outcomes in which Sam and Ruby each roll the same number and thus the difference between the numbers rolled is 0 .

For the remaining $36-6=30$ possible outcomes, it is equally probable that the number on Ruby's roll is greater than the number on Sam's roll as it is that the number on Sam's roll is greater than the number on Ruby's roll.

That is, one half of these 30 , or 15 possible outcomes have a result that is negative and 15 have a result that is positive.

Therefore, the probability that the result from subtracting the number on Sam's roll from the number on Ruby's roll is negative is $\frac{15}{36}=\frac{5}{12}$.

ANSWER: (B)