To determine if events A (late arrival) and B (transportation by car) are independent, we need to find P(A), P(B), and then check if P(A) * P(B) equals P(A and B).
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Calculate P(A):
- Total late arrivals = 150
- Total arrivals = 450 \[ P(A) = \frac{\text{Total late arrivals}}{\text{Total arrivals}} = \frac{150}{450} = \frac{1}{3} \approx 0.3333 \]
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Calculate P(B):
- Total cars = 250
- Total arrivals = 450 \[ P(B) = \frac{\text{Total cars}}{\text{Total arrivals}} = \frac{250}{450} = \frac{5}{9} \approx 0.5556 \]
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Calculate P(A and B):
- Late arrivals by car = 20 \[ P(A \text{ and } B) = \frac{\text{Late arrivals by car}}{\text{Total arrivals}} = \frac{20}{450} = \frac{2}{45} \approx 0.0444 \]
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Check independence:
- For events A and B to be independent, the following must hold: \[ P(A) \times P(B) = P(A \text{ and } B) \]
First, we calculate \( P(A) \times P(B) \): \[ P(A) \times P(B) \approx 0.3333 \times 0.5556 \approx 0.1852 \]
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Conclusion: Now we check if \( P(A) \times P(B) \) (approximately 0.1852) equals \( P(A \text{ and } B) \) (approximately 0.0444).
Since these values are not equal, events A and B are not independent, but for the purpose of your question regarding the product:
The product \( P(A) \times P(B) \approx 0.1852 \) can be rounded to 0.19 when expressed as a decimal to the nearest hundredth.
Thus, the answer is: \[ \boxed{0.19} \]