To find \( P(\text{yellow or glossy}) \), we can use the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Let:
- \( A \) be the event that a finish is yellow.
- \( B \) be the event that a finish is glossy.
Step 1: Calculate \( P(A) \)
The total number of finishes is 30. The number of yellow finishes is 7 (both matte and glossy).
\[ P(A) = \frac{\text{Number of yellow finishes}}{\text{Total finishes}} = \frac{7}{30} \]
Step 2: Calculate \( P(B) \)
To find \( P(B) \), we need to find the total number of glossy finishes. The total number of glossy finishes is 20 (as shown in the table).
\[ P(B) = \frac{\text{Number of glossy finishes}}{\text{Total finishes}} = \frac{20}{30} \]
Step 3: Calculate \( P(A \cap B) \)
The number of yellow glossy finishes is 7.
\[ P(A \cap B) = \frac{\text{Number of yellow glossy finishes}}{\text{Total finishes}} = \frac{7}{30} \]
Step 4: Combine the probabilities
Now we can calculate \( P(A \cup B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{7}{30} + \frac{20}{30} - \frac{7}{30} \]
Simplifying this:
\[ P(A \cup B) = \frac{7 + 20 - 7}{30} = \frac{20}{30} \]
Thus, the probability \( P(\text{yellow or glossy}) \) is:
\[ \boxed{\frac{20}{30}} \]