If $$P(A \cup B) = \frac{5}{9}, \quad P(\overline{A} \cup \overline{B}) = \frac{13}{27}, \quad P(A) = \frac{11}{18},$$ then find the odds against the event $$B$$.

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**Solution:**

1. Given:
   - $$P(A \cup B) = \frac{5}{9}$$
   - $$P(\overline{A} \cup \overline{B}) = \frac{13}{27}$$
   - $$P(A) = \frac{11}{18}$$

2. Recall the identity:
   $$P(\overline{A} \cup \overline{B}) = 1 - P(A \cap B)$$

3. From the union formula:
   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

4. Calculate $$P(A \cap B)$$:
   $$P(A \cap B) = 1 - P(\overline{A} \cup \overline{B}) = 1 - \frac{13}{27} = \frac{14}{27}$$

5. Substitute values into the union formula:
   $$
   \frac{5}{9} = \frac{11}{18} + P(B) - \frac{14}{27}
   $$

6. Find common denominators and simplify:
   - Convert all to denominator 54:
     - $$\frac{5}{9} = \frac{30}{54}$$
     - $$\frac{11}{18} = \frac{33}{54}$$
     - $$\frac{14}{27} = \frac{28}{54}$$

So,
   $$\frac{30}{54} = \frac{33}{54} + P(B) - \frac{28}{54}$$

7. Simplify:
   $$\frac{30}{54} = \frac{5}{54} + P(B)$$

8. Solve for $$P(B)$$:
   $$P(B) = \frac{30}{54} - \frac{5}{54} = \frac{25}{54}$$

9. Odds against event $$B$$ are given by:
   $$\text{Odds against } B = \frac{P(\overline{B})}{P(B)} = \frac{1 - P(B)}{P(B)} = \frac{1 - \frac{25}{54}}{\frac{25}{54}} = \frac{\frac{29}{54}}{\frac{25}{54}} = \frac{29}{25}$$

**Answer:** $$\frac{29}{25}$$

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