Rita the painter rolls a fair $6\text{-sided die}$that has $3$ red sides, $2$ yellow sides, and $1$ blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple?
To find the probability that Rita mixes the color purple, we need to determine the probability that she rolls a red side and a blue side in two rolls of the die. 

1. **Identify the probabilities of each color:**
   - Probability of rolling a red side ($P(R)$): There are 3 red sides out of 6, so $P(R) = \frac{3}{6} = \frac{1}{2}$.
   - Probability of rolling a blue side ($P(B)$): There is 1 blue side out of 6, so $P(B) = \frac{1}{6}$.

2. **Calculate the probability of rolling a red side first and a blue side second:**
   \[
   P(R \text{ first and } B \text{ second}) = P(R) \times P(B) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}
   \]

3. **Calculate the probability of rolling a blue side first and a red side second:**
   \[
   P(B \text{ first and } R \text{ second}) = P(B) \times P(R) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}
   \]

4. **Add the probabilities of the two mutually exclusive events:**
   \[
   P(\text{Red and Blue in any order}) = P(R \text{ first and } B \text{ second}) + P(B \text{ first and } R \text{ second}) = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6}
   \]

Thus, the probability that Rita mixes the color purple is $\frac{1}{6}$.

The final answer is $\boxed{\frac{1}{6}}$