Question

a and b are two events. let ( p(a)=0.3 ), ( p(b)=0.9 ), and ( p(a \text{ and } b)=0.24 ). which statement is true? ( 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)=p(a) ) and ( p(b|a)=p(b) ). ( a ) and ( b ) are independent events because ( p(a|b)=p(a) ) and ( p(b|a)=p(b) ). ( a ) and ( b ) are not independent events because ( p(a|b)=p(b) ) and ( p(b|a)=p(a) )

Explanation:

Step1: Calculate \(P(A|B)\)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).
Substitute \(P(A\cap B) = 0.24\) and \(P(B)=0.9\) into the formula:
\(P(A|B)=\frac{0.24}{0.9}=\frac{24}{90}=\frac{4}{15}\approx0.27\)

Step2: Compare \(P(A|B)\) with \(P(A)\)

Given \(P(A) = 0.3\).
Since \(P(A|B)\approx0.27
eq0.3 = P(A)\)

Answer:

A. \(A\) and \(B\) are not independent events because \(P(A|B)
eq P(A)\)