Question

what is the right anwser
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 check if the product \( P(A) \times P(B) \) equals \( P(A \cap B) \).

First, let’s find the probabilities:

1. **Calculate \( P(A) \)**:
- Total number of arrivals: 450
- Number of late arrivals (Event A): 150
- \( P(A) = \frac{\text{Number of Late Arrivals}}{\text{Total Arrivals}} = \frac{150}{450} = \frac{1}{3} \approx 0.3333 \)

2. **Calculate \( P(B) \)**:
- Total number of arrivals: 450
- Number of arrivals by car (Event B): 250
- \( P(B) = \frac{\text{Number of Arrivals by Car}}{\text{Total Arrivals}} = \frac{250}{450} = \frac{5}{9} \approx 0.5556 \)

3. **Calculate \( P(A \cap B) \)**:
- Number of late arrivals by car: 20
- \( P(A \cap B) = \frac{\text{Late Arrivals by Car}}{\text{Total Arrivals}} = \frac{20}{450} = \frac{2}{45} \approx 0.0444 \)

Now, to check for independence, \( P(A) \times P(B) \) needs to equal \( P(A \cap B) \).

4. **Calculate \( P(A) \times P(B) \)**:
\[
P(A) \times P(B) = \left(\frac{1}{3}\right) \times \left(\frac{5}{9}\right) = \frac{5}{27} \approx 0.1852
\]

Since we want the product \( P(A) \times P(B) \) to be equal to `P(A ∩ B)` (which is approximately 0.0444) for independence, we express the answer as \( P(A) \times P(B) \) and round to the nearest hundredth:

\[
\text{Final answer:} \approx 0.19
\]

Thus, the product \( P(A) \times P(B) \) should equal **0.19** for the events to be independent.