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Alejandro surveyed his classmates to determine who has ever gone surfing and who has ever gone snowboarding.

Let [tex]\(A\)[/tex] be the event that the person has gone surfing, and let [tex]\(B\)[/tex] be the event that the person has gone snowboarding.

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
& \text{Has Snowboarded} & \text{Never Snowboarded} & \text{Total} \\
\hline
\text{Has Surfed} & 36 & 189 & 225 \\
\hline
\text{Never Surfed} & 12 & 63 & 75 \\
\hline
\text{Total} & 48 & 252 & 300 \\
\hline
\end{tabular}
\][/tex]

Which statement is true about whether [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent events?

A. [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent events because [tex]\(P(A \mid B) = P(A) = 0.16\)[/tex].

B. [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent events because [tex]\(P(A \mid B) = P(A) = 0.75\)[/tex].

C. [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are not independent events because [tex]\(P(A \mid B) = 0.16\)[/tex] and [tex]\(P(A) = 0.75\)[/tex].

D. [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are not independent events because [tex]\(P(A \mid B) = 0.75\)[/tex] and [tex]\(P(A) = 0.16\)[/tex].

Sagot :

GinnyAnswer
To determine whether the events [tex]\(A\)[/tex] (the person has gone surfing) and [tex]\(B\)[/tex] (the person has gone snowboarding) are independent, we need to compare [tex]\(P(A)\)[/tex] and [tex]\(P(A \mid B)\)[/tex].

Given the data:
- Total number of people surveyed: [tex]\(300\)[/tex]
- Number of people who have gone surfing ([tex]\(A\)[/tex]): [tex]\(225\)[/tex]
- 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]

First, let’s compute the probability [tex]\(P(A)\)[/tex]:
[tex]\[ P(A) = \frac{\text{Number of people who have gone surfing}}{\text{Total number of people surveyed}} = \frac{225}{300} = 0.75 \][/tex]

Next, let’s compute the probability [tex]\(P(B)\)[/tex]:
[tex]\[ P(B) = \frac{\text{Number of people who have gone snowboarding}}{\text{Total number of people surveyed}} = \frac{48}{300} = 0.16 \][/tex]

Next, let’s compute the probability [tex]\(P(A \cap B)\)[/tex]:
[tex]\[ P(A \cap B) = \frac{\text{Number of people who have gone both surfing and snowboarding}}{\text{Total number of people surveyed}} = \frac{36}{300} = 0.12 \][/tex]

Now, we compute the conditional probability [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]

To check for independence, we compare [tex]\(P(A)\)[/tex] and [tex]\(P(A \mid B)\)[/tex]. If [tex]\(P(A) = P(A \mid B)\)[/tex], then the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.

We have:
[tex]\[ P(A) = 0.75 \][/tex]
[tex]\[ P(A \mid B) = 0.75 \][/tex]

Since [tex]\(P(A) = P(A \mid B)\)[/tex], the events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.

Therefore, the correct statement is:
[tex]\[ \text{\(A\) and \(B\) are independent events because \(P(A \mid B) = P(A) = 0.75\).} \][/tex]

So, the true statement here is:
[tex]\[ \boxed{A \text{ and } B \text{ are independent events because } P(A \mid B) = P(A) = 0.75.} \][/tex]
This corresponds to the second choice in the given options.
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