To determine whether the events [tex]\( A \)[/tex] (a person has gone surfing) and [tex]\( B \)[/tex] (a person has gone snowboarding) are independent, we need to analyze their probabilities and see if they satisfy the condition for independence: [tex]\( P(A \mid B) = P(A) \)[/tex].
Let’s examine the data and calculate the required probabilities step-by-step.
### Step 1: Extract the Totals
Given the table, we have:
- Total number of people surveyed: [tex]\(300\)[/tex]
- Number of people who have surfed ([tex]\(A\)[/tex]): [tex]\(225\)[/tex]
- Number of people who have snowboarded ([tex]\(B\)[/tex]): [tex]\(48\)[/tex]
- Number of people who have both surfed and snowboarded ([tex]\(A \cap B\)[/tex]): [tex]\(36\)[/tex]
### Step 2: Calculate [tex]\( P(A) \)[/tex]
The probability that a person has gone surfing, [tex]\( P(A) \)[/tex], is the ratio of the number of people who have surfed to the total number of people:
[tex]\[
P(A) = \frac{\text{Number of people who have surfed}}{\text{Total number of people}} = \frac{225}{300} = 0.75
\][/tex]
### Step 3: Calculate [tex]\( P(B) \)[/tex]
The probability that a person has gone snowboarding, [tex]\( P(B) \)[/tex], is:
[tex]\[
P(B) = \frac{\text{Number of people who have snowboarded}}{\text{Total number of people}} = \frac{48}{300} = 0.16
\][/tex]
### Step 4: Calculate [tex]\( P(A \cap B) \)[/tex]
The probability that a person has both surfed and snowboarded, [tex]\( P(A \cap B) \)[/tex], is:
[tex]\[
P(A \cap B) = \frac{\text{Number of people who have both surfed and snowboarded}}{\text{Total number of people}} = \frac{36}{300} = 0.12
\][/tex]
### Step 5: Calculate [tex]\( P(A \mid B) \)[/tex]
The conditional probability that a person has surfed given that they have snowboarded, [tex]\( P(A \mid B) \)[/tex], is:
[tex]\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.16} = 0.75
\][/tex]
### Step 6: Compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex]
We have:
- [tex]\( P(A \mid B) = 0.75 \)[/tex]
- [tex]\( P(A) = 0.75 \)[/tex]
Since [tex]\( P(A \mid B) = P(A) \)[/tex], the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
### Conclusion
The correct statement is:
[tex]\[ A \text{ and } B \text{ are independent events because } P(A \mid B) = P(A) = 0.75. \][/tex]
