Based on long experience, an airline has found that about 6% of the people making reservations on a flight from Miami to Denver do not show up for the flight. Suppose the airline overbooks this flight by selling 267 ticket reservations for an airplane with only 255 seats. Let n = 267 represent the number of ticket reservations. Let r represent the number of people with reservation s who show up for the flight. Which expression represents the probability that a seat will be available for everyone who shows up holding a reservation? $P(255 \leq r)$; $P(r \leq 255)$; $P(r \leq 267)$; $P(r=255)$

Air South overbooks as many as five passengers per flight because some passengers with reservations do not show up for the flight. The airline’s records indicate the following probabilities that it will be overbooked at flight time:

$$\begin{matrix} \text{Number overbooked at flight time} & \text{0} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5}\\ \text{Probability of number overbooked} & \text{0.75} & \text{0.10} & \text{0.06} & \text{0.04} & \text{0.04} & \text{0.01}\\ \end{matrix}$$

Find the expected number over for each flight.

Find each integral. $\int \csc ^{2} \pi t d t$

An airline finds that 5% of the persons making reservations on a certain flight will not show up for the flight. If the airline sells 160 tickets for a flight that has only 155 seats, what is the probability that a seat will be available for every person holding a reservation and planning to fly?

Airlines often sell more tickets for a flight than there are seats, because some ticket holders don’t show up for the flight. Assume that an airplane has $100$ seats for passengers and that the probability that a person holding a ticket appears for the flight is $0.90$. If the airline sells $105$ tickets, what is the probability that everyone who appears for the flight will get a seat?

Airlines often oversell seats on an airplane. This is done so that the seats for the few passengers that are no-shows will still have been sold. Sometimes, however, all passengers show up and there are more ticketed passengers than seats. Suppose that one flight has 160 passengers and only 156 seats. Determine the number of ways that the airline can select 4 people at random to place on a different flight.

In order to avoid empty seats, airlines generally sell more tickets than there are seats. Suppose that the number of ticketed passengers who show up for a flight with 100 seats is a normal random variable with mean 97 and standard deviation 6. What is the probability that at least one person will have to be “bumped” (denied space on the plane)?

A small airline overbooks flights on the assumption that several passengers will not show up. Suppose that the probability that a passenger shows up is 0.91. What is the probability that a 20-seat flight with 22 tickets sold will be able to seat all passengers who arrive?

An airline knows that, on average, the probability that a passenger will not show up for a flight is 6%. If an airplane is fully booked and holds 300 passengers, how many seats are expected to be empty? If the airline overbooked the flight by 10 passengers, about how many passengers are expected to show up for the flight? Justify your answer.

An airline makes 200 reservations for a flight that holds 185 passengers. The probability that a passenger arrives for the flight is 0.9, and the passengers are assumed to be independent. a. Approximate the probability that all the passengers who arrive can be seated. b. Approximate the probability that the flight has empty seats. c. Approximate the number of reservations that the airline should allow so that the probability that everyone who arrives can be seated is 0.95. [Hint: Successively try values for the number of reservations.]

Airline overbooking. An airline knows that over the long run, 90% of passengers who reserve seats show up for their flight. On a particular flight with 300 seats, the airline accepts 324 reservations.

a) Assuming that passengers show up independently of each other, what is the chance that the flight will be overbooked?

b) Suppose that people tend to travel in groups. Would that increase or decrease the probability of overbooking? Explain your answer.

c) Redo the calculation a) assuming that passengers always travel in pairs. Check that your answers to a), b), and $c$ ) are consistent.

As reported by a spokesperson for Southwest Airlines, the no-show rate for reservations is 16 percent. In other words, the probability is 0.16 that a person making a reservation will not take the flight. For the next flight, 42 people have reservations. What is the probability that

a. exactly 5 do not take the flight?

b. between 9 and 12 , inclusive, do not take the flight?

c. at least 1 does not take the flight?

d. at most 2 do not take the flight?

e. Comment on the accuracy of the normal approximation in this case.

Because all airline passengers do not show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10 , and the passengers behave independently.

What is the probability that the flight departs with empty seats?

Because all airline passengers do not show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently. a. What is the probability that every passenger who shows up can take the flight? b. What is the probability that the flight departs with empty seats?