Question

the following two - way table shows the distribution of high school students categorized by their grade level and book - type preference.

book - type preference

	fiction	nonfiction	total
sophomore	26	13	39
total	46	35	81

suppose a high school student is selected at random. let event a = junior and event b = fiction. are events a and b independent?

yes, p(a)=p(b|a)

no, p(a)≠p(b|a)

no, p(a)≠p(a|b)

yes, p(a)=p(a|b)

Explanation:

Step1: Calculate \(P(A)\)

Let \(A\) be some event (not clearly defined in terms of grade - level here, assume \(A\) is being a junior). \(P(A)=\frac{42}{81}\) since there are 42 juniors out of 81 total students.

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

If \(B\) is the event of preferring fiction. \(P(A|B)=\frac{20}{46}\) because among the 46 students who prefer fiction, 20 are juniors.
Since \(\frac{42}{81}
eq\frac{20}{46}\), we check the independence condition. For two events \(A\) and \(B\) to be independent, \(P(A) = P(A|B)\).

Answer:

No, \(P(A)
eq P(A|B)\)