Question

belleville high school offers classes on three different foreign languages. let a be the event that a student is in eleventh grade, and let b be the event that a student is enrolled in french class. which statement is true about whether a and b are independent events? a and b are independent events because ( p(a|b) ) ( = p(a) ). a and b are independent events because ( p(a|b) ) ( = p(b) ). a and b are not independent events because ( p(a|b) ) (
eq p(a) ). a and b are not independent events because ( p(a|b) ) (
eq p(b) ).

Explanation:

Step1: Calculate \(P(A)\)

The total number of students is \(n = 552\). The number of students in eleventh - grade (\(A\)) is \(n(A)=138\). So \(P(A)=\frac{n(A)}{n}=\frac{138}{552}=\frac{1}{4}\)

Step2: Calculate \(P(A|B)\)

The number of students in French class (\(B\)) is \(n(B) = 272\). The number of students who are in eleventh - grade and French class is \(n(A\cap B)=68\). By the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)}\), we have \(P(A|B)=\frac{68}{272}=\frac{1}{4}\)

Since \(P(A|B) = P(A)\), events \(A\) and \(B\) are independent.

Answer:

A and B are independent events because \(P(A|B)=P(A)\)