91Ó°ÊÓ

We know that events \(A\) and \(B\) are independent, meaning that the occurrence of one event doesn't affect the occurrence of the other event. We are given the probabilities: \(P(A) = 0.20\) and \(P(B) = 0.15\). To calculate \(P(A \cup B)\) for independent events, we can use the following formula: \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\)

Since events A and B are independent, we can compute probabilities of their complements as follows: 1. Probability of not pre-purchasing gas: \(P(A') = 1 - P(A) = 1 - 0.20 = 0.80\) 2. Probability of not renting a GPS: \(P(B') = 1 - P(B) = 1 - 0.15 = 0.85\)

To construct the table, we will use the probabilities derived in Step 2 and multiply each with the hypothetical 1000 cases. The table will look like this: | | Gas Pre-Purchased (A) | Gas Not Pre-Purchased (A') | Total | |----------------|-----------------------|----------------------------|-------| | GPS Rented (B) | 30 | 120 | 150 | | No GPS (B') | 170 | 680 | 850 | | Total | 200 | 800 | 1000 | The values in the table are calculated using the probabilities and the hypothetical 1000 cases, e.g., -GPS rented (B) and gas pre-purchased (A): \(1000 \times P(A) \times P(B) = 1000 \times 0.20 \times 0.15 = 30\) -GPS rented (B) and gas not pre-purchased (A'): \(1000 \times P(A') \times P(B) = 1000 \times 0.80 \times 0.15 = 120\) -And other values are calculated in a similar manner.

Using the values from the table, we can calculate the probability of the union of events A and B as follows: \(P(A \cup B) = \frac{30 + 120 + 170}{1000} = \frac{320}{1000} = 0.32\) The long-run relative frequency interpretation of this probability is that, in the long run, for 1000 car rentals, about 32% (or 320 out of 1000 times) the renter would either pre-purchase gas or rent a GPS device or both.