Question

Alli rolls a standard -sided die twice. What is the probability of rolling integers that differ by on her first two rolls?

Answer 1

We have to use a little bit of casework to solve this problem because some numbers on the die have a positive difference of 2 when paired with either of two other numbers (for example, 3 with either 1 or 5) while other numbers will only have a positive difference of 2 when paired with one particular number (for example, 2 with 4).

If the first roll is a 1, 2, 5, or 6, there is only one second roll in each case that will satisfy the given condition, so there are 4 combinations of rolls that result in two integers with a positive difference of 2 in this case. If, however, the first roll is a 3 or a 4, in each case there will be two rolls that satisfy the given condition- 1 or 5 and 2 or 6, respectively. This gives us another 4 successful combinations for a total of 8.

Since there are 6 possible outcomes when a die is rolled, there are a total of
`6\cdot6=36` possible combinations for two rolls, which means our probability is
`$(8)/(36)=\boxed{(2)/(9)}.$`

OR

We can also solve this problem by listing all the ways in which the two rolls have a positive difference of 2:

(6,4), (5,3), (4,2), (3,1), (4,6), (3,5), (2,4), (1,3).

So, we have 8 successful outcomes out of
`$6\cdot 6 = 36$` possibilities, which produces a probability of
`(8)/(36) = \frac29.`

Answer 2

okay lol i lol i i lol lol lol just said that he’s not