Question

a survey was taken of children between the ages of 7 and 12. let a be the event that the person rides the bus to school, and let b be the event that the person has 3 or more siblings.

	0 siblings	1 sibling	2 siblings	3 or more siblings	total
bikes to school	8	9	8	2	27
rides bus to school	18	36	12	9	75
is driven to school	32	58	22	10	122
total	82	140	54	24	300

which statement is true about whether a and b are independent events?
a and b are independent events because p(a | b) = p(a) = 0.12.
a and b are independent events because p(a | b) = p(a) = 0.25.
a and b are not independent events because p(a | b) = 0.12 and p(a) = 0.25.
a and b are not independent events because p(a | b) = 0.375 and p(a) = 0.25.

Explanation:

Step1: Calculate \(P(A)\)

The total number of children surveyed is \(n = 300\). The number of children who ride the bus to school (event \(A\)) is \(75\). So \(P(A)=\frac{75}{300}=0.25\)

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

The number of children with 3 or more siblings (event \(B\)) is \(n(B)=24\). The number of children with 3 or more siblings who ride the bus to school is \(9\). So \(P(A|B)=\frac{9}{24}=0.375\)

Step3: Check independence

Two events \(A\) and \(B\) are independent if \(P(A|B)=P(A)\). Since \(P(A|B) = 0.375\) and \(P(A)=0.25\), \(A\) and \(B\) are not independent events.

Answer:

A and B are not independent events because \(P(A|B)=0.375\) and \(P(A) = 0.25\)