Question

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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 0.09, 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 338 tickets, for a plane that has a maximum capacity of 311 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. Your final answer for the problem should have four decimal digits after the decimal point.

Answer 1

Given

A customer cancels his/her ticket in the last minute with probability = 0.09

maximum capacity = 311 seats

if the airline sells = 338 tickets

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 - 0.09 = 0.91

Average number of people in the flight if 338 tickets are booked

= 338 * 0.91= 307.58 =

Variance of the number of people in the flight if 338 tickets are booked

= 338 * 0.91 * 0.09 = 27.6822

Standard deviation of the number of people in the flight if 338 tickets are booked

=
The flight will be of over capacity if more than 311 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

P(X > 311)
=

= P( Z > 0.6500)

= 0.2578

Hence, the probability that the flight will be of over capacity if '338' tickets are booked = 0.2578

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