Expert: Josh - 2/26/2008

Question
I have done a 30 week table on a set of traffic lights on the probability of what colour they would be on three days of each of the 30 weeks when I arrived at them. The lights are green for 30 seconds, amber for 10 seconds and red for twenty seconds. I made a board with a circle cut into these times & colours & used a spinning top as the idicator. Now I have to choose a week & work out what the theoretical probability of the lights being different to those recorded during my simulation. I have chosen week ten where the lights were green on Monday, green on Tuesday and red on Wednesday. Could you please help me work out the method of how I would go about this & the result. I am thinking 3 colours over 90 days = 1/30 am I anywhere near correct? Thank you

Answer
Majella,

It would be helpful if you told me whether you are solving this problem at a secondary or tertiary level. There are different ways of going about this, in regard to methodology and whether your observed data is in fact representative of the actual process -- the changing patterns of a set of traffic lights.

It is unclear whether the "green (30 sec), amber (10 sec), red (20s)" part refers to your observation on a particular day, or the aggregated total for a week or more. I think the first question you need to sort out is how you compute the probabilities. (Again, the appropriate explanation depends on how much you know)

1. If you are an uni student, I could talk about probability density functions, such as a normal distribution, which would allow for variation about the mean duration of each traffic light color)

2. If this is a high school project, we would approach the problem differently. We can consider the ensemble average. i.e., assuming the processes are independent and identically distributed, you can add up the total duration of each traffic light color (red,amber and green) for the entire observation period (the 90 days) and divide each by the total number of minutes accumulated in this observation period. This would give you the probability of each color as P(X=green), P(X=amber), P(X=red). And of course, the three figures add up to 1.

Now, we focus on a much narrower observation window. We take a snap shot on a particular week. Again, the actual task you mentioned is subject to interpretation. You really need to ask the person who posed the question to find out exactly what he/she means. As I understand it, for the week 10 sample, we have "Green, Green, Red". If we base our probability estimates on this very limited sample set, we would arrive at P(X=Green)=2/3, P(X=Red)=1/3 and P(X=amber)=0. One question you may ask is how much does this estimate DEVIATE from the ensemble average obtained using the data gathered over a 90 day period. Is this a biased estimate?

Another view would be this: On Monday in week 10, the light is green. Based on your extensive study, what is the probability that the light is anything but green. Using your ensemble averages, you can answer this by calculating P(X=amber)+P(X=red).

How you draw such comparisons depends on what assumptions you make. Before you conduct this study, i.e., without knowledge of your observed data, a fair guess would be that P(red), P(amber) and P(green) are equally probable. i.e., P(red)=P(amber)=P(green)=1/3. You should check with your teacher and make sure you solve the problem the way it was intended. We can make guesses all day, but the instructions should be a lot clearer.