Answer :
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 [tex]\(P(A)\)[/tex] and [tex]\(P(A \mid B)\)[/tex].
Given the data:
- Total number of people surveyed: [tex]\(300\)[/tex]
- Number of people who have gone surfing ([tex]\(A\)[/tex]): [tex]\(225\)[/tex]
- Number of people who have gone snowboarding ([tex]\(B\)[/tex]): [tex]\(48\)[/tex]
- Number of people who have gone both surfing and snowboarding ([tex]\(A \cap B\)[/tex]): [tex]\(36\)[/tex]
First, let’s compute the probability [tex]\(P(A)\)[/tex]:
[tex]\[ P(A) = \frac{\text{Number of people who have gone surfing}}{\text{Total number of people surveyed}} = \frac{225}{300} = 0.75 \][/tex]
Next, let’s compute the probability [tex]\(P(B)\)[/tex]:
[tex]\[ P(B) = \frac{\text{Number of people who have gone snowboarding}}{\text{Total number of people surveyed}} = \frac{48}{300} = 0.16 \][/tex]
Next, let’s compute the probability [tex]\(P(A \cap B)\)[/tex]:
[tex]\[ P(A \cap B) = \frac{\text{Number of people who have gone both surfing and snowboarding}}{\text{Total number of people surveyed}} = \frac{36}{300} = 0.12 \][/tex]
Now, we compute the conditional probability [tex]\(P(A \mid B)\)[/tex]:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.16} = 0.75 \][/tex]
To check for independence, we compare [tex]\(P(A)\)[/tex] and [tex]\(P(A \mid B)\)[/tex]. If [tex]\(P(A) = P(A \mid B)\)[/tex], then the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.
We have:
[tex]\[ P(A) = 0.75 \][/tex]
[tex]\[ P(A \mid B) = 0.75 \][/tex]
Since [tex]\(P(A) = P(A \mid B)\)[/tex], the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] 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]
So, the true statement here is:
[tex]\[ \boxed{A \text{ and } B \text{ are independent events because } P(A \mid B) = P(A) = 0.75.} \][/tex]
This corresponds to the second choice in the given options.
Given the data:
- Total number of people surveyed: [tex]\(300\)[/tex]
- Number of people who have gone surfing ([tex]\(A\)[/tex]): [tex]\(225\)[/tex]
- Number of people who have gone snowboarding ([tex]\(B\)[/tex]): [tex]\(48\)[/tex]
- Number of people who have gone both surfing and snowboarding ([tex]\(A \cap B\)[/tex]): [tex]\(36\)[/tex]
First, let’s compute the probability [tex]\(P(A)\)[/tex]:
[tex]\[ P(A) = \frac{\text{Number of people who have gone surfing}}{\text{Total number of people surveyed}} = \frac{225}{300} = 0.75 \][/tex]
Next, let’s compute the probability [tex]\(P(B)\)[/tex]:
[tex]\[ P(B) = \frac{\text{Number of people who have gone snowboarding}}{\text{Total number of people surveyed}} = \frac{48}{300} = 0.16 \][/tex]
Next, let’s compute the probability [tex]\(P(A \cap B)\)[/tex]:
[tex]\[ P(A \cap B) = \frac{\text{Number of people who have gone both surfing and snowboarding}}{\text{Total number of people surveyed}} = \frac{36}{300} = 0.12 \][/tex]
Now, we compute the conditional probability [tex]\(P(A \mid B)\)[/tex]:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.16} = 0.75 \][/tex]
To check for independence, we compare [tex]\(P(A)\)[/tex] and [tex]\(P(A \mid B)\)[/tex]. If [tex]\(P(A) = P(A \mid B)\)[/tex], then the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.
We have:
[tex]\[ P(A) = 0.75 \][/tex]
[tex]\[ P(A \mid B) = 0.75 \][/tex]
Since [tex]\(P(A) = P(A \mid B)\)[/tex], the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] 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]
So, the true statement here is:
[tex]\[ \boxed{A \text{ and } B \text{ are independent events because } P(A \mid B) = P(A) = 0.75.} \][/tex]
This corresponds to the second choice in the given options.
