Question

Small aircraft arrive at a private airport according to a given Poisson process at a rate of 6 per hour.

1. What is the probability that exactly 5 small aircraft arrive in the first hour (after opening) and exactly 7 arrive in the hour after that?
2. What is the probability that fewer than 5 small aircraft arrive in the first hour and at least 10 arrive in the hour after that?
3. What is the probability that exactly 5 small aircraft arrive in the first hour and exactly 12 aircraft arrive in the first two hours?

Answer 1

The arrival rate for the small aircrafts per hour here is given as 6 per hour. Therefore the arrival process here could be modelled as:

a) The probability that exactly 5 small aircraft arrive in the first hour (after opening) and exactly 7 arrive in the hour after that is computed here as:

P(X = 5) P(X = 7)

This is computed using poisson probability function here as:

Therefore 0.0221 is the required probability here.

b) The probability that fewer than 5 small aircraft arrive in the first hour and at least 10 arrive in the hour after that is computed here as:
P(X < 5)P(X >= 10) = P(X <= 4)*(1 - P(X <= 9))

This is computed in EXCEL here as:
=poisson.dist(4,6,TRUE)*(1-poisson.dist(9,6,TRUE))

0.0239 is the output here.

Therefore 0.0239 is the required probability here.

c) The probability that exactly 5 small aircraft arrive in the first hour and exactly 12 aircraft arrive in the first two hours

= The probability that exactly 5 small aircraft arrive in the first hour and exactly (12 - 5) = 7 aircraft arrive in the next hour

This is same as part a)

Therefore 0.0221 is the required probability here.