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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 the probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] with the probability of [tex]\( A \)[/tex].

First, let's find the individual probabilities:

1. Probability of A:
[tex]\( P(A) \)[/tex] is the probability that a randomly chosen student has gone surfing.

[tex]\[ P(A) = \frac{\text{Number of students who have gone surfing}}{\text{Total number of students}} = \frac{225}{300} = 0.75 \][/tex]

2. Probability of B:
[tex]\( P(B) \)[/tex] is the probability that a randomly chosen student has gone snowboarding.

[tex]\[ P(B) = \frac{\text{Number of students who have gone snowboarding}}{\text{Total number of students}} = \frac{48}{300} = 0.16 \][/tex]

3. Conditional Probability [tex]\( P(A \mid B) \)[/tex]:
[tex]\( P(A \mid B) \)[/tex] is the probability of having gone surfing given that the student has gone snowboarding.

[tex]\[ P(A \mid B) = \frac{\text{Number of students who have gone surfing and snowboarding}}{\text{Number of students who have gone snowboarding}} = \frac{36}{48} = 0.75 \][/tex]

To check if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex].

Given:
[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.

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