To determine the likelihood that a student is in the anime club given that they take Japanese, we will use the concept of conditional probability. Conditional probability is the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred, and it is calculated using the formula:
[tex]\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Here, [tex]\( A \)[/tex] represents the event that a student is in the anime club and [tex]\( B \)[/tex] represents the event that a student takes Japanese. We need to find [tex]\( P(A|B) \)[/tex], which is the probability that a student is in the anime club given that they take Japanese.
From the table provided:
- [tex]\( P(A \cap B) \)[/tex] is the probability that a student is both in the anime club and takes Japanese. According to the table, this probability is 0.15.
- [tex]\( P(B) \)[/tex] is the probability that a student takes Japanese. According to the table, this probability is 0.20.
Using the conditional probability formula:
[tex]\[ P(\text{in anime club}|\text{takes Japanese}) = \frac{P(\text{in anime club and takes Japanese})}{P(\text{takes Japanese})} \][/tex]
Substituting the values from the table:
[tex]\[ P(\text{in anime club}|\text{takes Japanese}) = \frac{0.15}{0.20} \][/tex]
This quotient yields:
[tex]\[ P(\text{in anime club}|\text{takes Japanese}) = 0.75 \][/tex]
Thus, given that a student takes Japanese, the likelihood that he or she is in the anime club is [tex]\( 0.75 \)[/tex] or 75%.
