Question

Suppose that the number of vehicles passing by the NUS entrance A is modelled by a homogeneous Poisson process with the rate of 3 vehicles per minute. Each arriving vehicle is either a lorry or a car. It is a lorry with probability 0.2 and it is a car with probability 0.8. In addition, the types of the arriving vehicles are independent.

(a) (3 pts) Find the probability that in 3 mins, at least one lorry passes by the NUS entrance A.
(b) (3 pts) Suppose that during the first 15 minute of observation, exactly 12 lorries have passed by. Find the expected number of the cars passing the entrance during the same period.
(c) (4 pts) Suppose that during the first minute of observation, exactly 15 vehicles have passed by. Find the probability that among these vehicles, there were exactly 3 lorries and 12 cars.

Answer 1

a) The probability that at least one lorry passes by in 3 minutes is 0.488. b) The expected number of cars passing the entrance in the first 15 minutes is 9. c) The probability of exactly 3 lorries and 12 cars passing by in the first minute is (15 choose 3) * (0.2)³ * (0.8)¹²

Step-by-step explanation:

(a) To find the probability that at least one lorry passes by the NUS entrance A in 3 minutes, we can use the complement rule. The probability that no lorries pass by in 3 minutes is given by:

P(no lorries in 3 mins) = (0.8)³ = 0.512

So, the probability that at least one lorry passes by is:

P(at least one lorry in 3 mins) = 1 - P(no lorries in 3 mins) = 1 - 0.512 = 0.488

(b) Since the number of lorries passing by follows a Poisson process with rate 0.2 per minute, the expected number of lorries passing by in 15 minutes is:

Expected number of lorries = rate x time = 0.2 x 15 = 3

Therefore, the expected number of cars passing the entrance in the same period would be:

Expected number of cars = total number of vehicles - expected number of lorries = 12 - 3 = 9

(c) To find the probability of exactly 3 lorries and 12 cars among the 15 vehicles passing by in the first minute, we can use the probability mass function of the binomial distribution. The probability is given by:

P(3 lorries and 12 cars) = (15 choose 3) * (0.2)³ * (0.8)¹²