To determine the correct answers for the statement, let's break down the information given in the table.
1. Identify the given probabilities and definitions:
- Total number of people surveyed: 200
- Number of people who do not hike regularly: 100
- Number of people who live near a lake: 48
- Number of people who do not hike regularly and live near a lake: 24
2. Calculate the required probabilities:
- Probability of not regularly hiking (P(Not Hike)):
[tex]\[
P(\text{Not Hike}) = \frac{\text{Number of people who do not hike regularly}}{\text{Total number of people}} = \frac{100}{200} = 0.5
\][/tex]
- Probability of living near a lake (P(Lake)):
[tex]\[
P(\text{Lake}) = \frac{\text{Number of people who live near a lake}}{\text{Total number of people}} = \frac{48}{200} = 0.24
\][/tex]
- Conditional probability of not regularly hiking given that a person lives near a lake (P(Not Hike | Lake)):
[tex]\[
P(\text{Not Hike | Lake}) = \frac{\text{Number of people who do not hike regularly and live near a lake}}{\text{Number of people who live near a lake}} = \frac{24}{48} = 0.5
\][/tex]
3. Determine if the events are independent:
- Two events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if:
[tex]\[
P(A | B) = P(A)
\][/tex]
Here, we need to check if [tex]\(P(\text{Not Hike | Lake})\)[/tex] equals [tex]\(P(\text{Not Hike})\)[/tex].
- As calculated:
[tex]\[
P(\text{Not Hike | Lake}) = 0.5 \quad \text{and} \quad P(\text{Not Hike}) = 0.5
\][/tex]
- Since [tex]\(P(\text{Not Hike | Lake}) = P(\text{Not Hike})\)[/tex], the events "not regularly hiking" and "living near a lake" are independent.
Now we can complete the statement:
Not regularly hiking and living near a lake are independent events because the probability of not regularly hiking given that a person lives near a lake is equal to the probability of not regularly hiking.
