To calculate \( P(\text{yellow or glossy}) \), we can use the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
where:
- \( A \) is the event of selecting a yellow item,
- \( B \) is the event of selecting a glossy item.
Step 1: Calculate \( P(A) \)
The number of yellow items is 7 (from the table). The total number of items is 30.
\[ P(A) = \frac{\text{Number of yellow items}}{\text{Total number of items}} = \frac{7}{30} \]
Step 2: Calculate \( P(B) \)
To find the number of glossy items, we sum the glossy finishes for each color:
- Red glossy: 9
- Blue glossy: 4
- Yellow glossy: 7
Total glossy items:
\[ 9 + 4 + 7 = 20 \]
Thus, the probability of selecting a glossy item is:
\[ P(B) = \frac{\text{Number of glossy items}}{\text{Total number of items}} = \frac{20}{30} \]
Step 3: Calculate \( P(A \cap B) \)
\( A \cap B \) is the event of selecting a yellow item that is also glossy. From the table, there are 7 glossy yellow items.
Thus,
\[ P(A \cap B) = \frac{\text{Number of yellow glossy items}}{\text{Total number of items}} = \frac{7}{30} \]
Step 4: Calculate \( P(A \cup B) \)
Now we can substitute into our union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Plugging in the values:
\[ P(A \cup B) = \frac{7}{30} + \frac{20}{30} - \frac{7}{30} \]
Simplifying:
\[ P(A \cup B) = \frac{7 + 20 - 7}{30} = \frac{20}{30} \]
Therefore, the probability \( P(\text{yellow or glossy}) \) is
\[ \frac{20}{30} \]
In fraction form and without reducing, the final answer is
\[ \frac{20}{30} \]
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