1. The problem asks us to multiply the fractions $\frac{1}{3}$ and $\frac{1}{2}$.\n\n2. The formula for multiplying fractions is:\n$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$\nwhere $a$, $b$, $c$, and $d$ are integers and $b, d \neq 0$.\n\n3. Applying the formula to our fractions:\n$$\frac{1}{3} \times \frac{1}{2} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}$$\n\n4. There are no common factors to simplify in $\frac{1}{6}$, so this is the final answer.\n\nTherefore, the product of $\frac{1}{3}$ and $\frac{1}{2}$ is $\frac{1}{6}$.