Introduction
Upon a trip home, my mother and I were noticing a very peculiar occurrence: Traffic lights were almost continuously green. Indeed, exactly fifteen different traffic lights were green consecutively. Now, I am bad at probability, but this seems unlikely.
Probability Application
I reasoned that since there are three different options for all fifteen traffic lights, the probability of fifteen traffic lights being consecutively green was one in $_{15}P_{3}$. This is because there is only one sequence of traffic light configurations where all of them are green and we must count all of the possible traffic light configurations using a permutation since a sequence such as G,R,Y is not the same as Y,R,G. With that said, the probability of the event comes out to be approximately $0.0366\%$.
Question
Is this application of probability correct, incorrect, or somewhere in the middle? To make this question very precise, 'somewhere in the middle' means that my application of probability makes many underlying assumptions and ignores many factors. I am not aware what these specific assumptions and factors may be (if they exist), but that's why I ask this question. In what sense am I correct, and in what sense am I incorrect?
Best Answer
This is a deliberate design feature of many urban traffic control systems. For an explanation see this Wikipedia article about the "green wave".