Detailed Solution:
Step 1: Concept of Dilution and Replacement
When a fraction of a liquid is replaced repeatedly, the amount of the original liquid left after \( n \) repetitions is given by:
\[ \text{Remaining Fraction of Original Liquid} = \left( 1 – \text{Fraction Replaced} \right)^n \]
Step 2: Apply the formula
Fraction replaced = \( \frac{2}{3} \), so the remaining fraction = \( 1 – \frac{2}{3} = \frac{1}{3} \).
Number of repetitions, \( n = 4 \).
\[ \text{Fraction of Milk Remaining} = \left( \frac{1}{3} \right)^4 = \frac{1}{81} \]
Step 3: Fraction of water
Since the total content in the glass is always 1:
\[ \text{Fraction of Water} = 1 – \text{Fraction of Milk Remaining} = 1 – \frac{1}{81} = \frac{80}{81} \]
Step 4: Final Ratio of Milk to Water
\[ \text{Milk : Water} = \frac{1}{81} : \frac{80}{81} = 1 : 80 \]
Final Answer: The final ratio of milk to water is \( \boxed{1 : 80} \).