Question

Color Matte Finish Glossy Finish TOTAL Red 8 9 17 Blue 2 4 6 Yellow 0 7 7 TOTAL 10 20 30 What is P(yellow or glossy) ? Provide the answer in fraction form. You do not need to reduce your answer.

Answer 1

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

\[
\frac{20}{30}
\]