Question

Suppose that the probability that a passenger will miss a flight is 0.0934. Airlines do not like flights with empty seats, but it is also not desirable to have overbooked flights because passengers must be "bumped" from the flight. Suppose that an airplane has a seating capacity of 58 passengers.

(a) If 60 tickets are sold, what is the probability that 59 or 60 passengers show up for the flight resulting in an overbooked flight?
(b) Suppose that 64 tickets are sold. What is the probability that a passenger will have to be "bumped"?
(c) For a plane with seating capacity of 53 passengers, how many tickets may be sold to keep the probability of a passenger
(a) The probability of an overbooked flight is .02 (Round to four decimal places as needed.)
(b) The probability that a passenger will have to be bumped is (Round to four decimal places as needed.)

Answer 1

To find the probability of an overbooked flight and the probability of a passenger being bumped, we use the binomial probability formula. By calculating the probabilities for different scenarios, we can determine the likelihood of these events occurring. The number of tickets that may be sold to keep a certain probability of a passenger being bumped can be determined by finding the smallest value of k that satisfies the condition.

Step-by-step explanation:

(a) To find the probability that 59 or 60 passengers show up for the flight resulting in an overbooked flight, we can use the binomial probability formula. The probability of exactly 59 passengers showing up is given by P(X = 59) = C(60, 59) * (0.0934)^59 * (1 - 0.0934)^(60-59), and the probability of exactly 60 passengers showing up is given by P(X = 60) = C(60, 60) * (0.0934)^60 * (1 - 0.0934)^(60-60). The probability of an overbooked flight is the sum of these two probabilities, P(X ≥ 59) = P(X = 59) + P(X = 60).

(b) To find the probability that a passenger will have to be 'bumped' when 64 tickets are sold, we can find the probability of exactly 64 passengers showing up, and subtract it from 1 to find the probability of at least 65 passengers showing up. The probability of exactly 64 passengers showing up is given by P(X = 64) = C(64, 64) * (0.0934)^64 * (1 - 0.0934)^(64-64). The probability of at least 65 passengers showing up is 1 - P(X ≤ 64) = 1 - (P(X = 0) + P(X = 1) + ... + P(X = 64)).

(c) To find the number of tickets that may be sold for a plane with seating capacity of 53 passengers to keep the probability of a passenger being bumped equal to 0.02, we can find the smallest value of k such that the probability of at least k passengers showing up is less than or equal to 0.02. We can use the same approach as in part (b) to calculate this probability. Starting with k = 0, we calculate P(X ≤ k) until we find a value of k that satisfies the condition.