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.

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$$\begin{tabular}{|c|c|c|c|c|c|} \\hline & 0 siblings & 1 sibling & 2 siblings & 3 or more siblings & total \\\\ \\hline walks to school & 24 & 37 & 12 & 3 & 76 \\\\ \\hline bikes to school & 8 & 9 & 8 & 2 & 27 \\\\ \\hline rides bus to school & 18 & 36 & 12 & 9 & 75 \\\\ \\hline is driven to school & 32 & 58 & 22 & 10 & 122 \\\\ \\hline total & 82 & 140 & 54 & 24 & 300 \\\\ \\hline \\end{tabular}$$

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:

Define the events and extract values

Using the Two-Way Frequency Tables knowledge point

• Total number of children surveyed: \(N = 300\)
• Event \(A\): Rides the bus to school. Total for this row is \(75\).
• Event \(B\): Has 3 or more siblings. Total for this column is \(24\).
• Intersection \(A \cap B\): Rides the bus and has 3 or more siblings. The cell value is \(9\).

Calculate the probability of event A

Using the Conditional Probability Calculation knowledge point

$$ P(A) = \frac{\text{Total who ride the bus}}{\text{Total surveyed}} = \frac{75}{300} = 0.25 $$

Calculate the conditional probability P(A|B)

Using the Conditional Probability Calculation knowledge point

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{9}{24} = 0.375 $$

Determine independence

Using the Independent Events knowledge point

• Two events are independent if and only if \(P(A|B) = P(A)\).
• Since \(P(A|B) = 0.375\) and \(P(A) = 0.25\), we have \(P(A|B)

eq P(A)\).

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

Answer:

• 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 \(P(A) = 0.25\) (Correct answer)