For independent events \( X \) and \( Y \), the probability of at least one occurring is given by:
\[ P(X \cup Y) = P(X) + P(Y) - P(X)P(Y). \]
Substituting the given values:
\[ 0.8 = \frac{1}{3} + n - \left(\frac{1}{3}\right)n. \]
Simplifying:
\[ 0.8 = \frac{1}{3} + n - \frac{n}{3}. \]
Combining terms:
\[ 0.8 = \frac{1}{3} + \frac{2n}{3}. \]
Multiplying the entire equation by 3:
\[ 2.4 = 1 + 2n. \]
Rearranging:
\[ 2n = 1.4 \implies n = 0.7 = \frac{7}{10}. \]