Sure! To determine [tex]\( P \)[/tex] (pepperoni and olives), we'll use the formula that relates the probabilities of individual events with their intersection and union.
The formula is:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Where:
- [tex]\( P(A \cup B) \)[/tex] is the probability of either event [tex]\( A \)[/tex] or event [tex]\( B \)[/tex] occurring (which is given as 0.8 for either pepperoni or olives).
- [tex]\( P(A) \)[/tex] is the probability of event [tex]\( A \)[/tex] occurring (which is given as 0.7 for pepperoni).
- [tex]\( P(B) \)[/tex] is the probability of event [tex]\( B \)[/tex] occurring (which is given as 0.6 for olives).
- [tex]\( P(A \cap B) \)[/tex] is the probability of both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring (which we need to find for both pepperoni and olives).
To isolate [tex]\( P(A \cap B) \)[/tex], we can rearrange the formula:
[tex]\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \][/tex]
Substitute the given values into the formula:
[tex]\[ P(\text{pepperoni and olives}) = 0.7 + 0.6 - 0.8 \][/tex]
Perform the arithmetic:
[tex]\[ P(\text{pepperoni and olives}) = 1.3 - 0.8 \][/tex]
[tex]\[ P(\text{pepperoni and olives}) = 0.5 \][/tex]
Therefore, the probability that a customer likes both pepperoni and olives is [tex]\( 0.5 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{0.5} \][/tex]
