To determine whether events A (late arrival) and B (transportation by car) are independent, we need to calculate the probabilities \(P(A)\), \(P(B)\), and \(P(A \cap B)\).
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Calculate \(P(A)\): The total number of late arrivals is 150. The total number of arrivals is 450.
\[ P(A) = \frac{\text{Number of late arrivals}}{\text{Total number of arrivals}} = \frac{150}{450} = \frac{1}{3} \approx 0.333 \]
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Calculate \(P(B)\): The total number of arrivals by car is 250. The total number of arrivals is 450.
\[ P(B) = \frac{\text{Number of arrivals by car}}{\text{Total number of arrivals}} = \frac{250}{450} = \frac{5}{9} \approx 0.556 \]
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Calculate \(P(A \cap B)\): The number of late arrivals by car is 20. Thus,
\[ P(A \cap B) = \frac{\text{Number of late arrivals by car}}{\text{Total number of arrivals}} = \frac{20}{450} \approx 0.0444 \]
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Check for independence: Events A and B are independent if: \[ P(A \cap B) = P(A) \times P(B) \]
Now, calculate \(P(A) \times P(B)\):
\[ P(A) \times P(B) = \left(\frac{1}{3}\right) \times \left(\frac{5}{9}\right) = \frac{5}{27} \approx 0.1852 \]
To see if events A and B are independent, we compare \(P(A \cap B)\) to \(P(A) \times P(B)\):
- \(P(A \cap B) \approx 0.0444\)
- \(P(A) \times P(B) \approx 0.1852\)
Since \(P(A \cap B) \neq P(A) \times P(B)\), the events A and B are not independent.
However, to answer the original question about what the product \(P(A) \times P(B)\) needs to equal to show independence, it needs to equal \(P(A \cap B)\):
\[ \text{For independence, } P(A \cap B) \text{ should equal } P(A) \times P(B). \]
Thus, \(P(A) \times P(B) \) should be approximately 0.044 (to the nearest hundredth).
Final answer: \[ \approx 0.044 \]