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 hundredt

Answer 1

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 \(P(A \cap B)\).

1. **Calculate \(P(A)\)**:
The total number of late arrivals is 150.
The total number of arrivals is 450.

\[
P(A) = \frac{\text{Number of late arrivals}}{\text{Total number of arrivals}} = \frac{150}{450} = \frac{1}{3} \approx 0.333
\]

2. **Calculate \(P(B)\)**:
The total number of arrivals by car is 250.
The total number of arrivals is 450.

\[
P(B) = \frac{\text{Number of arrivals by car}}{\text{Total number of arrivals}} = \frac{250}{450} = \frac{5}{9} \approx 0.556
\]

3. **Calculate \(P(A \cap B)\)**:
The number of late arrivals by car is 20.
Thus,

\[
P(A \cap B) = \frac{\text{Number of late arrivals by car}}{\text{Total number of arrivals}} = \frac{20}{450} \approx 0.0444
\]

4. **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:
\[
\approx 0.044
\]