Vehicles arrive according to a Poisson process at a given signalized intersection approach with one roadway lane. Using loop detectors, vehicle volumes in 5-minute intervals were collected during 500 intervals. From these intervals, the average 5-minute flow rate was computed to be 20 vehicles every 5 minutes.

Question: Determine the probability that more than 3 vehicles arrive in a time period of 30 seconds during the green time of the traffic signal.

To calculate the probability, we can use the Poisson distribution. The Poisson process assumes that the number of arrivals in a fixed interval of time follows a Poisson distribution, where the average rate of arrivals is known. In this case, we have the average 5-minute flow rate of 20 vehicles.

To determine the rate of arrivals in 30 seconds, we need to adjust the average rate. Since the average rate is given per 5 minutes, we can calculate the rate per 30 seconds by dividing it by 10 (30 seconds is 1/10th of 5 minutes). Therefore, the adjusted average rate is 2 vehicles every 30 seconds.

Now, we can use the Poisson distribution formula to find the probability of more than 3 vehicles arriving in a 30-second period. The formula is:

P(X > 3) = 1 - P(X ≤ 3)

Using the Poisson distribution table or a calculator, we can find the probabilities for each value from 0 to 3 and sum them up. Subtracting this sum from 1 will give us the probability of more than 3 vehicles.

Learn more about Poisson distribution

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