# The Waiting Time Paradox, or, Why Is My Bus Always Late?

![rw-book-cover](https://readwise-assets.s3.amazonaws.com/static/images/article4.6bc1851654a0.png)

## Metadata
- Author: [[jakevdp.github.io]]
- Full Title: The Waiting Time Paradox, or, Why Is My Bus Always Late?
- Category: #articles
- URL: https://jakevdp.github.io/blog/2018/09/13/waiting-time-paradox/
- Archive: https://web-archive.alecodes.page/bookmarks?bf=1&search=&title=The%20Waiting%20Time%20Paradox%2C%20or%2C%20Why%20Is%20My%20Bus%20Always%20Late%3F
> [!tldr]
> The text discusses the waiting time paradox in the context of bus arrivals. It explains that while one might expect the average wait time for a bus that arrives every 10 minutes to be 5 minutes, under certain assumptions, the average waiting time can actually be 10 minutes. The text explores this paradox through simulations and probabilistic arguments, showing that the average waiting time aligns with the average interval between bus arrivals. It also discusses the use of Poisson processes to model bus arrival times and concludes that the waiting time paradox assumption may not hold true in real-world scenarios where bus arrival intervals are not exponentially distributed.

## Highlights
Briefly, the inspection paradox arises whenever the probability of observing a quantity is related to the quantity being observed. Allen gives one example of surveying university students about the average size of their classes. Though the school may truthfully advertise an average of 30 students per class, the average class size *as experienced by students* can be (and generally will be) much larger. The reason is that there are (of course) more students in the larger classes, and so you oversample large classes when computing the average experience of students. [View Highlight](https://read.readwise.io/read/01j5vf1r4qjgrnqa3dq94f19e5))

