Let's analyze the data and determine whether events A and B are independent.
First, we define the events:
- Event [tex]\( A \)[/tex]: The person has gone surfing.
- Event [tex]\( B \)[/tex]: The person has gone snowboarding.
We have the following data from the survey:
- Total number of surveyed people: [tex]\( 300 \)[/tex]
- Total number of people who have gone surfing [tex]\( |A| \)[/tex]: [tex]\( 225 \)[/tex]
- Total 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]
We need to calculate the following probabilities:
1. [tex]\( P(A) \)[/tex]: The probability that a person has gone surfing.
2. [tex]\( P(B) \)[/tex]: The probability that a person has gone snowboarding.
3. [tex]\( P(A \cap B) \)[/tex]: The probability that a person has gone both surfing and snowboarding.
4. [tex]\( P(A \mid B) \)[/tex]: The probability that a person has gone surfing given that they have gone snowboarding.
### Calculating Probabilities
1. Probability of having gone surfing [tex]\( P(A) \)[/tex]:
[tex]\[ P(A) = \frac{|A|}{\text{Total number of surveyed people}} = \frac{225}{300} = 0.75 \][/tex]
2. Probability of having gone snowboarding [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{|B|}{\text{Total number of surveyed people}} = \frac{48}{300} = 0.16 \][/tex]
3. Probability of having gone both surfing and snowboarding [tex]\( P(A \cap B) \)[/tex]:
[tex]\[ P(A \cap B) = \frac{|A \cap B|}{\text{Total number of surveyed people}} = \frac{36}{300} = 0.12 \][/tex]
4. Probability of having gone surfing given that they have gone snowboarding [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]
### Determining Independence
Two events A and B are independent if and only if:
[tex]\[ P(A \mid B) = P(A) \][/tex]
In our case:
[tex]\[ P(A \mid B) = 0.75 \][/tex]
[tex]\[ P(A) = 0.75 \][/tex]
Since [tex]\( P(A \mid B) = P(A) \)[/tex], we conclude that:
[tex]\[ A \text{ and } B \text{ are independent events} \][/tex]
### Conclusion
The correct statement is:
[tex]\[ \boxed{A \text{ and } B \text{ are independent events because } P(A \mid B) = P(A) = 0.75.} \][/tex]
