Have you ever wondered about why you sometimes feel like you’ve waited ages for the bus to come, or why you always feel like you’ve come across more red lights than green lights during a road trip? Well, these phenomenons can be explained with the inspection paradox.

This paradox relates to the fact that observing a renewal interval at time t gives an interval with average value larger than that of an average renewal interval. This is complicated, right? Well here is some simple examples:

Imagine that you’re waiting for a public transport. Buses and trains are suppo

sed to arrive at constant intervals, but in practice some intervals are longer than others. With your luck, you might think you are more likely to arrive during a short interval. And you’re wrong: a random arrival is more likely to fall in a long interval because, well, it’s longer.

Let’s say that the shortest interval is 3 minutes whilst the longest is 10 minutes. At random arrival, there is a higher chance to arrive in the 10 minute interval than the 3 minute interval. This means that more people are waiting during the longer intervals causing the average of intervals between the trains to be lower than the average waiting time of all the passengers creating an inspection paradox.

This can also explain traffic lights: due to the reason you have to stop at a red light, you spend more time on it than the green lights, making you think more about bumping into red lights. However, from the traffic lights perspective, there is 50% chance that you see a green light but you just don’t think about it since you spend less time on it than red lights.