Question

Use the table to answer the question.

Arrival Time
Transport Type On Time Late TOTAL
Car 230 20 250
Bicycle 70 130 200
TOTAL 300 150 450
Let event A
represent a late arrival and event B
represent transportation by car. What would the product of P(A)
and P(B)
need to equal to show that the events are independent? Express the answer as a decimal to the nearest hundredth.

(1 point)

Answer 1

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).

1. **Calculate P(A)**:
- Total late arrivals = 150
- Total arrivals = 450
\[
P(A) = \frac{\text{Total late arrivals}}{\text{Total arrivals}} = \frac{150}{450} = \frac{1}{3} \approx 0.3333
\]

2. **Calculate P(B)**:
- Total cars = 250
- Total arrivals = 450
\[
P(B) = \frac{\text{Total cars}}{\text{Total arrivals}} = \frac{250}{450} = \frac{5}{9} \approx 0.5556
\]

3. **Calculate P(A and B)**:
- Late arrivals by car = 20
\[
P(A \text{ and } B) = \frac{\text{Late arrivals by car}}{\text{Total arrivals}} = \frac{20}{450} = \frac{2}{45} \approx 0.0444
\]

4. **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 \( P(A) \times P(B) \):
\[
P(A) \times P(B) \approx 0.3333 \times 0.5556 \approx 0.1852
\]

5. **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:
\[
\boxed{0.19}
\]