Question

Air traffic control is essential for the safe landing and departure of plans. You are monitoring a particular airport and observe that the arrival rate of small planes follows a Poisson distribution with an average of 6 small planes arriving per hour. Use this information and answer questions 1 to 4.

1. You decided to spend an hour to observe how many planes arrive. What is the probability that you observe exactly 6 small planes’ arrival? (use 3 decimal places)

2. On average, how many small planes would you expect to observe during a three-hour observation?

3. What is the probability that you observed at least 3 arrivals of small planes in the first 30 minutes of your observation? (use 3 decimal places)

4. You have observed 3 arrivals of small planes in the first 45 minutes of your observation. What is the probability that you observe a total of 5 small plane arrivals plans in the one-hour observation? (use 3 decimal places)

Answer 1

1)

probability that you observe exactly 6 small planes’ arrival =e^-6*6⁶/6! =0.161

2)expected number of planes in three-hour observation =2*6=18

3)expected number of planes in 30 minute observation =6*30/60=3

probability that you observed at least 3 arrivals of small planes in the first 30 minutes of your observation

=P(X>=3)=1-P(X<=2) =1-(e^-3*3⁰/0!+e^-3*3¹/1!+e^-3*3²/2!)=0.577

4)

probability that you observe a total of 5 small plane arrivals plans in the one-hour observation =P(2 planes in next 15 minutes)

=e^-1.51.5²/2! =0.251