1. **State the problem:** Calculate $\frac{2}{3} \div 1 \frac{5}{6}$. 2. **Convert mixed number to improper fraction:** $1 \frac{5}{6} = \frac{6 \times 1 + 5}{6} = \frac{11}{6}$. 3. **Rewrite division as multiplication by reciprocal:** $\frac{2}{3} \div \frac{11}{6} = \frac{2}{3} \times \frac{6}{11}$. 4. **Multiply numerators and denominators:** $$\frac{2 \times 6}{3 \times 11} = \frac{12}{33}$$ 5. **Simplify the fraction:** $$\frac{\cancel{12}^{4} \times 3}{\cancel{33}^{11} \times 3} = \frac{4}{11}$$ 6. **Final answer:** $\boxed{\frac{4}{11}}$