Question

Airlines typically overbook flights. Assume the airline sold 55 tickets for a plane with 50 seats. Let Y = the number of ticketed passengers who actually show up for the flight, with the pmf given below: Y ply) 45 0.05 46 0.10 47 0.12 48 0.14 49 0.25 50 0.17 51 0.06 52 0.05 53 0.03 54 0.02 55 0.01 Sum 1.00 If you are the fourth person on the standby list (i.e., you will be 4th person to get on if there are any available seats after all ticketed passengers have been seated), what is the probability that you will be able to take the flight?

Answer 1

The probability that the fourth person on the standby list can take the flight is 89%, calculated by summing the probabilities of 51 or fewer ticketed passengers showing up.

Step-by-step explanation:

To calculate the probability that you will be able to take the flight as the fourth person on the standby list, we need to consider the number of ticketed passengers (Y) who do not show up such that there are at least four unoccupied seats. Since the plane has 50 seats and 55 passengers have been ticketed, we look for the probability that 51 or fewer passengers show up. Using the given probability mass function (pmf), we sum up the probabilities for Y being 45 through 51.

• Probability (Y ≤ 45) = 0.05
• Probability (Y ≤ 46) = Probability (Y = 45) + P(Y = 46) = 0.05 + 0.10 = 0.15
• Probability (Y ≤ 47) = 0.15 + P(Y = 47) = 0.15 + 0.12 = 0.27
• Probability (Y ≤ 48) = 0.27 + P(Y = 48) = 0.27 + 0.14 = 0.41
• Probability (Y ≤ 49) = 0.41 + P(Y = 49) = 0.41 + 0.25 = 0.66
• Probability (Y ≤ 50) = 0.66 + P(Y = 50) = 0.66 + 0.17 = 0.83
• Probability (Y ≤ 51) = 0.83 + P(Y = 51) = 0.83 + 0.06 = 0.89

Thus, the probability that you will be able to take the flight as the fourth person on the standby list is 0.89 or 89%.