Question

To a parking lot of a commercial plaza on a weekend day, an average of 40 vehicles arrive per hour.
a) What is the probability that the time between the arrival of cars that arrive one after the other (consecutive) is between 6 and 12 minutes?
b) What is the probability that the time between the arrival of cars arriving one after the other (consecutive) is greater than the average by one standard deviation?
c) What is the probability that the time between the arrival of the 5th and 7th cars is more than 9 minutes?
d) What is the 90th percentile for the time between arrivals between the 5th and 7th cars?

Answer 1

We are given the average rate of arrivals as 40 per hour which means 40/60 = 2/3 vehicle per minute. Therefore average waiting time for a vehicle to arrive is given as: 3/2 = 1.5 minute.

Therefore the waiting time for a vehicle could be modelled here as:

a) The required probability here is computed as:

Therefore 0.0180 is the required probability here.

b) The standard deviation of the exponential distribution is equal to its mean that is 1.5 minute in this case. Therefore the required probability here is computed as:

Therefore 0.1353 is the required probability here.

c) The average arrival time between 2 arrivals is computed as: 3/2 * 2 = 3 minutes.

Therefore the probability that the above time is more than 9 minutes is computed here as:

Therefore 0.0500 is the required probability here.

d) Let the 90th percentile time required here be K.

Then, we have here: P( t > k) = 0.1

Therefore 6.9078 minutes is the required time interval here.