I am trying to solve the task
"Often the number of passengers on a flight is less than the number of bookings for that flight. The airline therefore overbooks with the risk of financially compensating any excess passengers. Assume that the airline has a revenue of 300 euros for each passenger flying with it, and a loss of 500 euros for each excess person who has booked a seat. Furthermore, each person who has booked a seat appears independently for the flight with probability p = 0.95. How many seats would you have at one
Airbus A319 with S=124 seats
Airbus A380 with S=555 seats
sell to maximize expected profit?
I did it like this, only with my values for S=124. But something is wrong. I know, that the expected value is 130 Seats.
Central limit problem. Maximizing profit when overbooking flights.
Here is my calculation:
$$P \left (\frac{S_{n}-0.95n}{\sqrt{0.95n(1-0.95)}}\leq{\frac{124-0.95n}{\sqrt{0.95n(1-0.95)}}} \right )\geq{\frac{500}{800}}$$
with a=300, b=500, S=124, p=0.95
$$\frac{S_{n}-0.95n}{\sqrt{0.95n(1-0.95)}}=X_{n}$$
Chat GPT says the Z-Score of $$\frac{500}{800}$$ is 0.5, but I know something is wrong, because n has to be 130
EDIT:
I got a new Z Score. The Z Score is 0.32
My Solution is:
Airbus A319 with S=124 seats: The Airline should sell 130 Seats
Airbus A380 with S=555 seats: The Airline should sell 582 Seats
Can someone please help me? :)