Answer :
To determine whether the events [tex]\(A\)[/tex] (has gone surfing) and [tex]\(B\)[/tex] (has gone snowboarding) are independent, we need to compare the probabilities [tex]\(P(A \mid B)\)[/tex] and [tex]\(P(A)\)[/tex].
### Step-by-Step Solution:
1. Calculate [tex]\(P(A)\)[/tex]:
[tex]\(P(A)\)[/tex] is the probability that a person has gone surfing.
[tex]\[ P(A) = \frac{\text{Total number of people who have gone surfing}}{\text{Total number of people surveyed}} = \frac{225}{300} = 0.75 \][/tex]
2. Calculate [tex]\(P(B)\)[/tex]:
[tex]\(P(B)\)[/tex] is the probability that a person has gone snowboarding.
[tex]\[ P(B) = \frac{\text{Total number of people who have gone snowboarding}}{\text{Total number of people surveyed}} = \frac{48}{300} = 0.16 \][/tex]
3. Calculate [tex]\(P(A \mid B)\)[/tex]:
[tex]\(P(A \mid B)\)[/tex] is the probability that a person has gone surfing given that they have gone snowboarding.
[tex]\[ P(A \mid B) = \frac{\text{Number of people who have both surfed and snowboarded}}{\text{Total number of people who have gone snowboarding}} = \frac{36}{48} = 0.75 \][/tex]
4. Determine Independence:
We compare [tex]\(P(A \mid B)\)[/tex] with [tex]\(P(A)\)[/tex]. For the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] to be independent, [tex]\(P(A \mid B)\)[/tex] should equal [tex]\(P(A)\)[/tex].
[tex]\[ P(A \mid B) = 0.75 \quad \text{and} \quad P(A) = 0.75 \][/tex]
Since [tex]\(P(A \mid B) = P(A)\)[/tex], events [tex]\(A\)[/tex] (has gone surfing) and [tex]\(B\)[/tex] (has gone snowboarding) are independent.
Therefore, the correct statement is:
[tex]\[ \text{A and B are independent events because } P(A \mid B) = P(A) = 0.75. \][/tex]
### Step-by-Step Solution:
1. Calculate [tex]\(P(A)\)[/tex]:
[tex]\(P(A)\)[/tex] is the probability that a person has gone surfing.
[tex]\[ P(A) = \frac{\text{Total number of people who have gone surfing}}{\text{Total number of people surveyed}} = \frac{225}{300} = 0.75 \][/tex]
2. Calculate [tex]\(P(B)\)[/tex]:
[tex]\(P(B)\)[/tex] is the probability that a person has gone snowboarding.
[tex]\[ P(B) = \frac{\text{Total number of people who have gone snowboarding}}{\text{Total number of people surveyed}} = \frac{48}{300} = 0.16 \][/tex]
3. Calculate [tex]\(P(A \mid B)\)[/tex]:
[tex]\(P(A \mid B)\)[/tex] is the probability that a person has gone surfing given that they have gone snowboarding.
[tex]\[ P(A \mid B) = \frac{\text{Number of people who have both surfed and snowboarded}}{\text{Total number of people who have gone snowboarding}} = \frac{36}{48} = 0.75 \][/tex]
4. Determine Independence:
We compare [tex]\(P(A \mid B)\)[/tex] with [tex]\(P(A)\)[/tex]. For the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] to be independent, [tex]\(P(A \mid B)\)[/tex] should equal [tex]\(P(A)\)[/tex].
[tex]\[ P(A \mid B) = 0.75 \quad \text{and} \quad P(A) = 0.75 \][/tex]
Since [tex]\(P(A \mid B) = P(A)\)[/tex], events [tex]\(A\)[/tex] (has gone surfing) and [tex]\(B\)[/tex] (has gone snowboarding) are independent.
Therefore, the correct statement is:
[tex]\[ \text{A and B are independent events because } P(A \mid B) = P(A) = 0.75. \][/tex]
