Question

Suppose that past experience shows that about 11% of passengers who are scheduled to take a particular flight fail to show up. For this reason, airlines sometimes overbook flights, selling more tickets than they have seats, with the expectation that they will have some no shows. Suppose an airline uses a small jet with seating for 30 passengers on a regional route and assume that passengers are independent of each other in whether they show up for the flight. Suppose that the airline consistently sells 32 tickets for every one of these flights.A. Describe a random variable X. Explain carefally what distribution can be use and what are a are the values of the parameters? B. On average, how many passengers will be on each flight? C. Find the probability that at least 40 seats will be filled. D. How often will they have enough seats for all of the passengers who show up for the flight?

Answer 1

(a) The average number of passengers that will be on each flight is 28.8.

(b) Everyone will have a seat on about 84.4% of the flights.

Explanation:

The correct question is:

Suppose that past experience shows that about 10% of passengers who are schedule to take a particular flight fail to show up. For this reason, airlines sometimes overbook flights, selling more tickets than they have seats, with the expectation that they will have some no shows. Suppose an airline used a small jet with seating for 30 passengers on a regional route and assume that passengers are independent of each other in whether they show up for the flight. Suppose that the airline consistently sells 32 tickets for every one of these flights.

(a) On average, how many passengers will be on each flight?

(b) How often will they have enough seats for all of the passengers who show up for the flight?

Solution:

Let the random variable X measure the number of passengers (out of 32) who show up for a flight.

For each passenger there is a 90% chance of showing up, so X is a binomial random variable with n = 32 and p = 0.90.

(a)

The average of a binomial random variable is:

`\text{Average}=np`

Compute the average number of passengers that will be on each flight as follows:

`\text{Average}=np`

`=32* 0.90\\=28.8`

Thus, the average number of passengers that will be on each flight is 28.8.

(b)

To have enough seats for all of the passengers who show up for the flight, the value of X must be less than or equal to 30.

Compute the value of P (X ≤ 30) as follows:

P (X ≤ 30) = 1 - P (X > 30)

= 1 - P (X = 31) - P (X = 32)

`=1-[{32\choose 31}\ (0.90)^(31)\ (1-0.90)^(32-31)]-[{32\choose 32}\ (0.90)^(32)\ (1-0.90)^(32-32)]\\\\=1-0.122087-0.034337\\\\=0.843576\\\\\approx 0.844`

Thus, everyone will have a seat on about 84.4% of the flights.