Probability of 15 consecutive green lights

Question

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?

Answer 1

If red, yellow, and green were equally likely and independent, the probability of getting $15$ greens in a row would be $1$ in $3^{15}$ or about $1$ in $14$ million. The probability of getting $15$ of something in a row would be $1$ in $3^{14}$ or about $1$ in $5$ million. Actually, yellow usually occupies much less time than red or green. Again, if we figure the chance of red and green to be $\frac 12$, you would have $1$ in $2^{15}=32768$ of hitting $15$ greens in a row. I think this is a demonstration that they are not independent. We have a street in San Francisco that I routinely hit six or eight in a row (after maybe hitting a red and waiting for the green) by driving exactly the speed limit.

Answer 2

You're basing your calculations on the assumption that all traffic lights operate independently, which is not only not necessarily true, but also most likely not true. Many large streets in my city have this feature that if you pass a green light once you will pass all the other traffic lights green. To achieve this one just needs to calculate the average time that it takes a car to go from one traffic light to the other and offset the lights changing time by that amount.

Answer 3

This is a deliberate design feature of many urban traffic control systems. For an explanation see this Wikipedia article about the "green wave".