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Chapter 5: Problem 10

Trains headed for destination \(A\) arrive at the train station at 15 -minute intervals starting at 7 A.M., whereas trains headed for destination \(B\) arrive at 15 -minute intervals starting at 7: 05 A.M. (a) If a certain passenger arrives at the station at a time uniformly distributed between 7 and 8 A.M. and then gets on the first train that arrives, what proportion of time does he or she go to destination \(A ?\) (b) What if the passenger arrives at a time uniformly distributed between 7: 10 and 8: 10 A.M.?

Short Answer

Expert verified
(a) The proportion of time the passenger goes to destination A when arriving between 7:00 AM and 8:00 AM is 62.5%. (b) The proportion of time the passenger goes to destination A when arriving between 7:10 AM and 8:10 AM is 37.5%.

Step by step solution

01

Determine the Train Arrival Times Before 8 AM

Based on the information provided, let's list the arrival times for both destination A and destination B trains before 8 AM: Destination A: 7:00, 7:15, 7:30, 7:45 Destination B: 7:05, 7:20, 7:35, 7:50
02

Divide the Timeframes for (a) and (b) into 15-minute Intervals

(a) For the passenger arriving between 7:00 AM and 8:00 AM, we can divide this into four 15-minute intervals: 1. 7:00 - 7:15 2. 7:15 - 7:30 3. 7:30 - 7:45 4. 7:45 - 8:00 (b) For the passenger arriving between 7:10 AM and 8:10 AM, we can also divide this into four 15-minute intervals: 1. 7:10 - 7:25 2. 7:25 - 7:40 3. 7:40 - 7:55 4. 7:55 - 8:10
03

Determine the Probability of Taking a Train to Destination A for Each Time Interval

Now we will calculate the probability of taking a train to destination A for each time interval in (a) and (b): (a) Between 7:00 AM and 8:00 AM: 1. In the first interval, the probability of taking a train to destination A is 1 since no train arrives at destination B. 2. In the second interval, both A and B trains arrive, and the probability of taking a train to A is 0.5. 3. In the third interval, the probability of taking a train to destination A is also 0.5. 4. In the fourth interval, the probability of taking a train to destination A is also 0.5. (b) Between 7:10 AM and 8:10 AM: 1. In the first interval, the probability of taking a train to destination A is 0 since no train arrives at destination A. 2. In the second interval, both A and B trains arrive, and the probability of taking a train to A is 0.5. 3. In the third interval, the probability of taking a train to destination A is also 0.5. 4. In the fourth interval, the probability of taking a train to destination A is also 0.5.
04

Calculate the Overall Probability for Each Scenario

We will now calculate the overall probability for each scenario by taking the average of the probabilities for each 15-minute interval: (a) Between 7:00 AM and 8:00 AM: Overall Probability = (1 + 0.5 + 0.5 + 0.5) / 4 = 0.625 (b) Between 7:10 AM and 8:10 AM: Overall Probability = (0 + 0.5 + 0.5 + 0.5) / 4 = 0.375 Therefore, the proportions of time the passenger goes to destination A are: (a) 62.5% for arrival times between 7:00 AM and 8:00 AM (b) 37.5% for arrival times between 7:10 AM and 8:10 AM

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