To determine whether the events [tex]\( A \)[/tex] (a student has gone surfing) and [tex]\( B \)[/tex] (a student has gone snowboarding) are independent, we'll check if [tex]\( P(A \mid B) = P(A) \)[/tex].
Given data:
- Total number of students surveyed: [tex]\( 300 \)[/tex]
- Students who have gone surfing ([tex]\( A \)[/tex]): [tex]\( 225 \)[/tex]
- Students who have gone snowboarding ([tex]\( B \)[/tex]): [tex]\( 48 \)[/tex]
- Students who have gone both surfing and snowboarding ([tex]\( A \cap B \)[/tex]): [tex]\( 36 \)[/tex]
First, we'll calculate the probabilities:
1. Probability that a student has gone surfing [tex]\( P(A) \)[/tex]:
[tex]\[
P(A) = \frac{\text{Number of students who have gone surfing}}{\text{Total number of students}} = \frac{225}{300}
\][/tex]
2. Probability that a student has gone snowboarding [tex]\( P(B) \)[/tex]:
[tex]\[
P(B) = \frac{\text{Number of students who have gone snowboarding}}{\text{Total number of students}} = \frac{48}{300}
\][/tex]
3. Probability that a student has gone surfing given that they have gone snowboarding [tex]\( P(A \mid B) \)[/tex]:
[tex]\[
P(A \mid B) = \frac{\text{Number of students who have gone both surfing and snowboarding}}{\text{Number of students who have gone snowboarding}} = \frac{36}{48}
\][/tex]
Now, substituting the values:
1. Calculating [tex]\( P(A) \)[/tex]:
[tex]\[
P(A) = \frac{225}{300} = 0.75
\][/tex]
2. Calculating [tex]\( P(B) \)[/tex] (though not directly needed):
[tex]\[
P(B) = \frac{48}{300}
\][/tex]
3. Calculating [tex]\( P(A \mid B) \)[/tex]:
[tex]\[
P(A \mid B) = \frac{36}{48} = 0.75
\][/tex]
Since [tex]\( P(A \mid B) = P(A) = 0.75 \)[/tex], the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are indeed independent.
Thus, 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].
So, the correct answer is:
[tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events because [tex]\( P(A \mid B) = P(A) = 0.75 \)[/tex].
