Question

Dr. Strangelove owns a sports club. He observed that the number of participants at each session is on average 16 people. This number changes due to randomness. He assumed the arrival process is Poisson. Due to the capacity of the facility, each session is limited to 20 people. If more than 20 people come to a session, he needs to ask those overflowed gym members to go home, which gives them a very bad impression for his business.

1.                  What is the proportion of the people who are asked to go home because the session is full? (Hint: Check the slides.)

2.                  In order to reduce the proportion of people who are asked to go home, he decided to increase the number of sessions. This change can reduce the average number of participants at each session equal to 12. What do you expect to happen to the proportion of the people who are asked to go home because the session is full? Guess which answer is correct. (Just guess the answer. I do not grade this problem. Confirm whether your guess is correct or not after you solve Question 3.) Note that 12/16 = 75%

(1) The proportion is about 75% of the answer in Question 1.

(2) The proportion is higher than 75% of the answer in Question 1.

(3) The proportion is slightly less than 75% of the answer in Question 1.

(4) The proportion is much less than 75% of the answer in Question 1.

3.                  The increase in the number of sessions makes the average number of participants at each session equal to 12. What is the proportion of the people who are asked to go home because the session is full?

(Hint: 4/sqrt(3) = 2.3, and Phi(2.3) = 0.99. Don’t just state the answer. You should explain how you solved the problem.)

Answer 1

Excel was used to solve this and snapshot is attached below:

For solving using hand calculations and not Excel, following formulas are to be used:

Poission Probability Function (Probability of EXACTLY x occurences in the time-interval/session):

For calculating probability of x<=2 (say), we need to calculate f(0), f(1) and f(2) and add them up.

So, P(x<=2) = f(0) + f(1) + f(2)

=> P(x>=2) = 1 - P(x<=2)

Similarly we can do for x>=20. Excel is easier. So, I have used that where there is an option for stating that probability that needs to be calculated is cumulative or individualistic. I have used cumulative by specifying "1".