Question

On any given flight, the goal of an airline is to fill the plane as much as possible, without exceeding the capacity of the plane. In order to achieve this, the airlines routinely overbook their flights in consideration of last minute cancellations. We assume that a customer cancels his/her ticket in the last minute with probability 0.08, independent of the other customers. We also assume that the airline is not able to sell more tickets in order to replace the canceled ones. What is the probability that a particular flight will be over capacity if the airline sells 323 tickets, for a plane that has a maximum capacity of 298 seats? In solving this problem, use the Central Limit Theorem, and in particular, use the De Moivre-Laplace normal approximation to the binomial distribution (with 1/2 correction) and be very careful when you choose the boundaries for probability computation. You will also need to use the standard normal CDF table that is in the summary notes that was made available to you for use during the exams. Use this table precisely as follows: In using the standard normal CDF table, first compute the input argument for the standard normal CDF with your calculator, then round this input argument value to two decimal digits after the decimal point, and finally locate the entry in the table which corresponds to the rounded input value. If you need the value of the standard normal CDF for arguments larger than 3.49 (not available in the table), you can use 1.0000.

Answer 1

Using the central limit theorem and the Demoivre-Laplace approximation,  we can assume a normal approximation to the binomial distribution with mean = np and variance = np(1-p)

The probability that a person doesn't cancel his/her ticket in the last minute = 1-0.08 = 0.92

Average number of people in the flight if 300 tickets are booked = 323 * 0.92 = 297.16

Variance of the number of people in the flight if 300 tickets are booked = 323*0.92*0.08 = 23.7728

Standard deviation of the number of people in the flight if 300 tickets are booked

The flight will be of over capacity if more than 298 people arrive. Let X be the number of people arriving in the flight.

We want to find the probability that the flight will be of over capacity. That is, we want to find

Hence, the probability that the flight will be of over capacity if 323 tickets are booked = 0.4317

Thank You. Please Upvote.