To find \( P(\text{yellow or glossy}) \), we need to determine the number of outcomes that satisfy either condition (yellow or glossy) and then divide by the total number of outcomes.
-
Number of Yellow items:
From the table, we see that there are \( 0 \) matte yellow items and \( 7 \) glossy yellow items, giving a total of \( 7 \) yellow items. -
Number of Glossy items:
The glossy finishes are as follows:- Red: \( 9 \)
- Blue: \( 4 \)
- Yellow: \( 7 \)
Total glossy items: \[ 9 + 4 + 7 = 20 \]
-
Using the Inclusion-Exclusion Principle:
We directly find \( P(\text{yellow or glossy}) \) as: \[ P(\text{yellow or glossy}) = P(\text{yellow}) + P(\text{glossy}) - P(\text{yellow and glossy}) \]-
\( P(\text{yellow}) \): Total number of yellow items is \( 7 \).
-
Total outcomes = \( 30 \), so: \[ P(\text{yellow}) = \frac{7}{30} \]
-
\( P(\text{glossy}) \): Total number of glossy items is \( 20 \). \[ P(\text{glossy}) = \frac{20}{30} \]
-
\( P(\text{yellow and glossy}) \): The only yellow items that are glossy are \( 7 \). Thus, \[ P(\text{yellow and glossy}) = \frac{7}{30} \]
Now, substitute these values into our equation: \[ P(\text{yellow or glossy}) = \frac{7}{30} + \frac{20}{30} - \frac{7}{30} \] \[ P(\text{yellow or glossy}) = \frac{20}{30} \]
-
-
Final Answer:
Thus, the probability \( P(\text{yellow or glossy}) \) is: \[ \frac{20}{30} \]
This is your answer in fraction form, and you do not need to reduce it.
1 answer