Given P(A)=(1)/(2), P(B^c)=(5)/(6), and P(A B)=(5)/(12). Find P(A B) and P(A B^c).

$$P(B) = 1 - P(B^c)$$ $$P(B) = 1 - \frac{5}{6}$$ $$P(B) = \frac{6}{6} - \frac{5}{6}$$ $$P(B) = \frac{1}{6}$$

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Substitute the known values:

$$\frac{5}{12} = \frac{1}{2} + \frac{1}{6} - P(A \cap B)$$

Find a common denominator for the fractions on the right side:

$$\frac{5}{12} = \frac{3}{6} + \frac{1}{6} - P(A \cap B)$$ $$\frac{5}{12} = \frac{4}{6} - P(A \cap B)$$

Simplify $\frac{4}{6}$ to $\frac{2}{3}$:

$$\frac{5}{12} = \frac{2}{3} - P(A \cap B)$$

Rearrange to solve for $P(A \cap B)$:

$$P(A \cap B) = \frac{2}{3} - \frac{5}{12}$$

Find a common denominator for $\frac{2}{3}$ and $\frac{5}{12}$:

$$P(A \cap B) = \frac{8}{12} - \frac{5}{12}$$ $$P(A \cap B) = \frac{3}{12}$$ $$P(A \cap B) = \boxed{\frac{1}{4}}$$

The event $A \setminus B^c$ means that event $A$ occurs and event $B^c$ does not occur. If $B^c$ does not occur, it means event $B$ occurs.
Therefore, $A \setminus B^c$ is equivalent to $A \cap B$.

$$P(A \setminus B^c) = P(A \cap B)$$