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The table shows the results of a survey of 200 randomly selected people about where they live and whether they hike regularly.

\begin{tabular}{|l|l|l|l|l|}
\hline & Lives In a City & \begin{tabular}{l}
Lives Near \\
Mountains
\end{tabular} & \begin{tabular}{l}
Lives Near a \\
Lake
\end{tabular} & Total \\
\hline Hikes & 28 & 48 & 24 & 100 \\
\hline Does Not Hike & 56 & 20 & 24 & 100 \\
\hline Total & 84 & 68 & 48 & 200 \\
\hline
\end{tabular}

Complete the given statement.

Not regularly hiking and living near a lake are [tex]$\square$[/tex] events because the probability of not regularly hiking given that a person lives near a lake [tex]$\square$[/tex] is to the probability of [tex]$\square$[/tex]



Answer :

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.