# Why You’re Waiting for Your Bus More Than You Should

## The bus paradox or the inspection paradox

Have you ever found yourself waiting for a bus, glancing impatiently at your watch, wondering why it always seems to take longer than expected? You might think it’s just bad luck, but there’s a mathematical explanation behind this seemingly frustrating experience. This phenomenon is known as the Bus Paradox, and understanding it not only sheds light on your daily commute but also on broader concepts in probability and statistics.

## The Bus Paradox Explained

The Bus Paradox, also known as the Waiting Time Paradox, refers to the counterintuitive reality that passengers often experience longer wait times for buses than the average bus interval would suggest. To illustrate, consider a bus route where buses are scheduled to arrive every 10 minutes. You might reasonably expect that, on average, you would wait about 5 minutes for a bus if you arrive at the stop at a random time. However, in reality, you often wait longer than this average.

This paradox arises from the inherent nature of random processes and the distribution of waiting times. To understand why this happens, let’s dive into some key concepts.