Two 6-sided dice are rolled. What is the probability that the sum is odd and the number on one of the dice is a 5?

Explanation:
Let's look at the ways we can achieve an odd result. Instead of listing out all 36 different roles, let's do it this way - I'm going to assume one die is Red and the other is Black. For each number on the Red die (1, 2, 3, 4, 5, 6), we get six different possible roles (for the 6 different possible roles of the Black die). So we get:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
1
2
3
4
5
6
1
E
O
E
O
E
O
2
O
E
O
E
O
E
3
E
O
E
O
E
O
4
O
E
O
E
O
E
5
E
O
E
O
E
O
6
O
E
O
E
O
E
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
If we count the number of ways we can get an odd number, we get 18. There are 36 different roles we can get, so the probability of getting an odd role as:
18
36
=
1
2