To determine if the events [tex]\( A \)[/tex] (the person has gone surfing) and [tex]\( B \)[/tex] (the person has gone snowboarding) are independent, we need to analyze the probabilities involved.
Here is the step-by-step solution:
1. Total number of people surveyed:
- Total = 300
2. Number of people who have gone surfing:
- Number of people who have surfed = 225
3. Number of people who have gone snowboarding:
- Number of people who have snowboarded = 48
4. Number of people who have both surfed and snowboarded:
- Number of people who have both surfed and snowboarded = 36
5. Calculate [tex]\( P(A) \)[/tex]:
- [tex]\( P(A) \)[/tex] is the probability that a person has surfed.
- [tex]\( P(A) = \frac{\text{Number of people who have surfed}}{\text{Total number of people}} = \frac{225}{300} = 0.75 \)[/tex]
6. Calculate [tex]\( P(B) \)[/tex]:
- [tex]\( P(B) \)[/tex] is the probability that a person has snowboarded.
- [tex]\( P(B) = \frac{\text{Number of people who have snowboarded}}{\text{Total number of people}} = \frac{48}{300} = 0.16 \)[/tex]
7. Calculate [tex]\( P(A \mid B) \)[/tex]:
- [tex]\( P(A \mid B) \)[/tex] is the conditional probability that a person has surfed given that they have snowboarded.
- [tex]\( P(A \mid B) = \frac{\text{Number of people who have both surfed and snowboarded}}{\text{Number of people who have snowboarded}} = \frac{36}{48} = 0.75 \)[/tex]
8. Compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex]:
- For the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent, [tex]\( P(A \mid B) \)[/tex] must equal [tex]\( P(A) \)[/tex].
- From the calculations: [tex]\( P(A \mid B) = 0.75 \)[/tex] and [tex]\( P(A) = 0.75 \)[/tex]
Since [tex]\( P(A \mid B) \)[/tex] is indeed equal to [tex]\( P(A) \)[/tex], the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
Therefore, the correct statement is:
A and B are independent events because [tex]\( P(A \mid B ) = P(A) = 0.75 \)[/tex].
