To determine whether the events [tex]\( A \)[/tex] (the person has gone surfing) and [tex]\( B \)[/tex] (the person has gone snowboarding) are independent, we need to compare the probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] with the probability of [tex]\( A \)[/tex].
First, let's find the individual probabilities:
1. Probability of A:
[tex]\( P(A) \)[/tex] is the probability that a randomly chosen student has gone surfing.
[tex]\[
P(A) = \frac{\text{Number of students who have gone surfing}}{\text{Total number of students}} = \frac{225}{300} = 0.75
\][/tex]
2. Probability of B:
[tex]\( P(B) \)[/tex] is the probability that a randomly chosen student has gone snowboarding.
[tex]\[
P(B) = \frac{\text{Number of students who have gone snowboarding}}{\text{Total number of students}} = \frac{48}{300} = 0.16
\][/tex]
3. Conditional Probability [tex]\( P(A \mid B) \)[/tex]:
[tex]\( P(A \mid B) \)[/tex] is the probability of having gone surfing given that the student has gone snowboarding.
[tex]\[
P(A \mid B) = \frac{\text{Number of students who have gone surfing and snowboarding}}{\text{Number of students who have gone snowboarding}} = \frac{36}{48} = 0.75
\][/tex]
To check if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex].
Given:
[tex]\[ P(A \mid B) = 0.75 \][/tex]
[tex]\[ P(A) = 0.75 \][/tex]
Since [tex]\( P(A \mid B) = P(A) \)[/tex], the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
Therefore, the correct statement is:
[tex]$A$[/tex] and [tex]$B$[/tex] are independent events because [tex]$P(A \mid B)=P(A)=0.75$[/tex].
