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A survey asks teachers and students whether they would like the new school mascot to be a shark or a moose. This table shows the results.

\begin{tabular}{|l|c|c|c|}
\hline & Sharks & Moose & Total \\
\hline Students & 90 & 10 & 100 \\
\hline Teachers & 5 & 10 & 15 \\
\hline Total & 95 & 20 & 115 \\
\hline
\end{tabular}

A person is randomly selected from those surveyed. Are being a student and preferring "shark" independent events? Why or why not?

A. Yes, they are independent because [tex]$P($[/tex]student[tex]$) \approx 0.87$[/tex] and [tex]$P($[/tex]student [tex]$\mid$[/tex] shark[tex]$) \approx 0.83$[/tex].

B. No, they are not independent because [tex]$P($[/tex]student[tex]$) \approx 0.87$[/tex] and [tex]$P($[/tex]student [tex]$\mid$[/tex] shark[tex]$) \approx 0.83$[/tex].

C. Yes, they are independent because [tex]$P($[/tex]student[tex]$) \approx 0.87$[/tex] and [tex]$P($[/tex]student [tex]$\mid$[/tex] shark[tex]$) \approx 0.95$[/tex].

D. No, they are not independent because [tex]$P($[/tex]student[tex]$) \approx 0.87$[/tex] and [tex]$P($[/tex]student [tex]$\mid$[/tex] shark[tex]$) \approx 0.95$[/tex].

Sagot :

GinnyAnswer
To determine whether being a student and preferring sharks are independent events, we need to compare the probability of being a student ([tex]\(P(\text{student})\)[/tex]) with the conditional probability of being a student given that the person prefers sharks ([tex]\(P(\text{student | shark})\)[/tex]).

1. Calculate [tex]\(P(\text{student})\)[/tex]:
[tex]\[ P(\text{student}) = \frac{\text{Number of students}}{\text{Total number of people surveyed}} = \frac{100}{115} \approx 0.87 \][/tex]

2. Calculate [tex]\(P(\text{student | shark})\)[/tex]:
From the table, there are 90 students who prefer sharks out of the total 95 people who prefer sharks.
[tex]\[ P(\text{student | shark}) = \frac{\text{Number of students who prefer sharks}}{\text{Total number of people who prefer sharks}} = \frac{90}{95} \approx 0.95 \][/tex]

3. Compare [tex]\(P(\text{student})\)[/tex] and [tex]\(P(\text{student | shark})\)[/tex]:
[tex]\[ P(\text{student}) \approx 0.87 \][/tex]
[tex]\[ P(\text{student | shark}) \approx 0.95 \][/tex]
Since [tex]\(P(\text{student})\)[/tex] is not equal to [tex]\(P(\text{student | shark})\)[/tex], the events are not independent.

Therefore, the correct answer is:
D. No, they are not independent because [tex]\(P(\text{student}) \approx 0.87\)[/tex] and [tex]\(P(\text{student | shark}) \approx 0.95\)[/tex].
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