Step 1: Convert every probability to a simple fraction first: $P(A) = \frac{1}{4}$, so $P(\bar{A}) = \frac{3}{4}$. Also $P(B|A) = \frac{1}{2}$ and $P(B|\bar{A}) = \frac{3}{4}$.
Step 2: Build a joint probability table using multiplication: $$P(A \cap B) = P(B|A)P(A) = \frac{1}{2}\times\frac{1}{4} = \frac{1}{8}$$ $$P(\bar{A} \cap B) = P(B|\bar{A})P(\bar{A}) = \frac{3}{4}\times\frac{3}{4} = \frac{9}{16}$$
Step 3: Add these two joint probabilities (since $A$ and $\bar{A}$ partition the sample space) to get the total probability of $B$: $$P(B) = \frac{1}{8} + \frac{9}{16} = \frac{2}{16} + \frac{9}{16} = \frac{11}{16}$$
Step 4: By definition of conditional probability, $$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{1/8}{11/16} = \frac{1}{8}\times\frac{16}{11} = \frac{16}{88} = \frac{2}{11}$$
Both routes (decimals or fractions) agree.
\[\boxed{P(A|B) = \frac{2}{11}}\]