Given events A and B are independent, we have the following properties:
1. P(A\ and\ B) = P(A) \cdot P(B)
2. P(B | A) = P(B)
3. P(A|B^C) = P(A)
4. P(A^C | B) = P(A^C)
Let's check the given conditions:
A) P(A\ and\ B) = 1/12
Using property 1, we have:
P(A\ and\ B) = P(A) \cdot P(B) = \dfrac{1}{3} \cdot \dfrac{1}{2} = \dfrac{1}{6} \neq \dfrac{1}{12}
B) P(B | A) = 1/3
Using property 2, we have:
P(B | A) = P(B) = \dfrac{1}{2} \neq \dfrac{1}{3}
C) P(A|B^C) = 2/3
Using property 3, we have:
P(A|B^C) = P(A) = \dfrac{1}{3} \neq \dfrac{2}{3}
D) P(A^C | B) = 2/3
Using property 4, we have:
P(A^C | B) = P(A^C) = 1 - P(A) = 1 - \dfrac{1}{3} = \dfrac{2}{3}
Therefore, the condition that could exist is:
\boxed{D) \ P(A^C | B) = \dfrac{2}{3}}