Question

question 8 of 10
a survey asks teachers and students whether they would like the new school
mascot to be a pirate or a moose. this table shows the results.

a person is randomly selected from those surveyed.
are being a student and preferring \pirate\ independent events? why or why
not?

a. no, they are not independent because ( p(\text{student}) approx 0.82 ) and
( p(\text{student|pirate}) approx 0.83 ).
b. no, they are not independent because ( p(\text{student}) approx 0.82 ) and
( p(\text{student|pirate}) approx 0.94 ).
c. yes, they are independent because ( p(\text{student}) approx 0.82 ) and
( p(\text{student|pirate}) approx 0.94 ).

Explanation:

Step1: Calculate \(P(\text{student})\)

The formula for probability is \(P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For \(P(\text{student})\), the number of students is \(90\) and the total number of people surveyed is \(110\). So \(P(\text{student})=\frac{90}{110}\approx0.82\).

Step2: Calculate \(P(\text{student}|\text{pirate})\)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A\) is “being a student” and \(B\) is “preferring pirate”.
\(P(A\cap B)\) (number of students who prefer pirate) is \(75\), and \(P(B)\) (number of people who prefer pirate) is \(80\). So \(P(\text{student}|\text{pirate})=\frac{75}{80}= 0.9375\approx0.94\).

Step3: Check for independence

Two events \(A\) and \(B\) are independent if \(P(A|B) = P(A)\).
Since \(P(\text{student})\approx0.82\) and \(P(\text{student}|\text{pirate})\approx0.94\), and \(0.82
eq0.94\), the events “being a student” and “preferring pirate” are not independent.

Answer:

B. No, they are not independent because \(P(\text{student})\approx0.82\) and \(P(\text{student}|\text{pirate})\approx0.94\).