To determine whether the events "red cover" and "fiction" are approximately independent, we need to use the concept of independent events in probability. Two events are independent if the probability of their intersection is equal to the product of their individual probabilities.
Let's define the events:
- [tex]\( P(\text{red cover}) = 0.25 \)[/tex]
- [tex]\( P(\text{fiction}) = 0.32 \)[/tex]
- [tex]\( P(\text{red cover and fiction}) = 0.08 \)[/tex]
For the events to be independent, the following condition must hold:
[tex]\[ P(\text{red cover and fiction}) = P(\text{red cover}) \times P(\text{fiction}) \][/tex]
Now, let's calculate the product of the individual probabilities:
[tex]\[ P(\text{red cover}) \times P(\text{fiction}) = 0.25 \times 0.32 = 0.08 \][/tex]
This matches the given probability of 0.08 for a book being both fiction and having a red cover:
[tex]\[ P(\text{red cover and fiction}) = 0.08 \][/tex]
Therefore, since:
[tex]\[ P(\text{red cover and fiction}) = P(\text{red cover}) \times P(\text{fiction}) \][/tex]
the events "red cover" and "fiction" are indeed approximately independent.
The correct justification is:
Approximately independent, because 8% = (25%) (32%)
So, the correct choice is:
- Approximately independent, because 8% = (25%) (32%)
