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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 if the events [tex]\( A \)[/tex] (the person has gone surfing) and [tex]\( B \)[/tex] (the person has gone snowboarding) are independent, we need to analyze the probabilities involved.

Here is the step-by-step solution:

1. Total number of people surveyed:
- Total = 300

2. Number of people who have gone surfing:
- Number of people who have surfed = 225

3. Number of people who have gone snowboarding:
- Number of people who have snowboarded = 48

4. Number of people who have both surfed and snowboarded:
- Number of people who have both surfed and snowboarded = 36

5. Calculate [tex]\( P(A) \)[/tex]:
- [tex]\( P(A) \)[/tex] is the probability that a person has surfed.
- [tex]\( P(A) = \frac{\text{Number of people who have surfed}}{\text{Total number of people}} = \frac{225}{300} = 0.75 \)[/tex]

6. Calculate [tex]\( P(B) \)[/tex]:
- [tex]\( P(B) \)[/tex] is the probability that a person has snowboarded.
- [tex]\( P(B) = \frac{\text{Number of people who have snowboarded}}{\text{Total number of people}} = \frac{48}{300} = 0.16 \)[/tex]

7. Calculate [tex]\( P(A \mid B) \)[/tex]:
- [tex]\( P(A \mid B) \)[/tex] is the conditional probability that a person has surfed given that they have snowboarded.
- [tex]\( P(A \mid B) = \frac{\text{Number of people who have both surfed and snowboarded}}{\text{Number of people who have snowboarded}} = \frac{36}{48} = 0.75 \)[/tex]

8. Compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex]:
- For the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent, [tex]\( P(A \mid B) \)[/tex] must equal [tex]\( P(A) \)[/tex].
- From the calculations: [tex]\( P(A \mid B) = 0.75 \)[/tex] and [tex]\( P(A) = 0.75 \)[/tex]

Since [tex]\( P(A \mid B) \)[/tex] is indeed equal to [tex]\( P(A) \)[/tex], the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.

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