Probability of Waiting Less Than 5 Minutes for a Bus: A Uniform Distribution Analysis

When considering bus schedules and passenger arrival times, one common problem is determining the probability that a passenger will wait less than a certain amount of time before catching a bus. In this article, we will explore the scenario where buses arrive at a specified stop with a 15-minute interval starting at 8:00 am, and a passenger arrives at a random time between 8:00 am and 8:30 am. We will use a uniform distribution to analyze the probability of the passenger waiting less than 5 minutes for a bus.

Bus Arrival Schedule

The bus schedule is as follows:

First bus: 8:00 am Second bus: 8:15 am Third bus: 8:30 am

Passenger Arrival Time

Assume the passenger arrives uniformly between 8:00 am and 8:30 am, which is a 30-minute time frame.

Waiting Time Analysis

The key is to determine the intervals during which the passenger can arrive and still wait less than 5 minutes for the next bus. We can break down the waiting times as follows:

For the 8:00 am bus, the passenger can arrive between 8:00 am and 8:05 am. For the 8:15 am bus, the passenger can arrive between 8:10 am and 8:15 am. For the 8:30 am bus, the passenger can arrive between 8:25 am and 8:30 am.

Each of these intervals is 5 minutes long.

Total Favorable Time Intervals

Summing these intervals gives us the total time where the passenger will wait less than 5 minutes for the bus:

8:00 am - 8:05 am: 5 minutes 8:10 am - 8:15 am: 5 minutes 8:25 am - 8:30 am: 5 minutes

Thus, the total favorable time is:

5 5 5 15 minutes

Probability Calculation

The total time frame in which the passenger can arrive is 30 minutes. Using the concept of uniform distribution, the probability of the passenger waiting less than 5 minutes for a bus is given by the ratio of the favorable outcomes to the total outcomes:

P(wait frac{15}{30} 0.5

Therefore, the probability that the passenger waits less than 5 minutes for a bus is 0.5 or 50%.

Conclusion

In conclusion, if a passenger arrives uniformly between 8:00 am and 8:30 am, the probability that they will wait less than 5 minutes for the next bus is 50%. This analysis demonstrates the power of understanding uniform distribution and its application in real-world scenarios such as bus schedules.