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 3?
There are a total of $6 \cdot 6 = 36$ possible outcomes when two 6-sided dice are rolled. We count the number of outcomes that satisfy the conditions: the sum is odd and the number on one of the dice is a 3.
For the sum to be odd, one of the dice must show an odd number and the other die must show an even number. So, we have two possibilities for which die will show 3. If the first die shows 3, then the second die must show an even number, which could be 2, 4, or 6. If the second die shows 3, then the first die must show an even number, which again could be 2, 4, or 6. Therefore, there are $2 \cdot 3 = 6$ possible outcomes that satisfy the conditions.
Hence, the probability that the sum is odd and the number on one of the dice is a 3 is $\dfrac{6}{36} = \boxed{\dfrac{1}{6}}$.