Question

An airline, believing that \(5 \%\) of passengers fail to show up for flights, overbooks (sells more tickets than there are seats). Suppose a plane will hold 265 passengers, and the airline sells 275 tickets. What's the probability the airline will not have enough seats, so someone gets bumped?

Short answer

The airline has a probability of approximately 15% of not having enough seats, causing someone to get bumped.

Step-by-step solution

Identify the Distribution

We assume the probability of a passenger not showing up is 0.05, and we have 275 tickets sold. This situation can be modeled by a binomial distribution where the number of trials is the number of tickets sold (275) and the probability of a 'success' (passenger not showing) is 0.05. Therefore, we define our variables as follows: \( n = 275 \) and \( p = 0.05 \).

Define the Complementary Event

To find the probability that the airline does not have enough seats, let's first calculate the probability that 265 or fewer passengers show up. Since we need at least one passenger to be bumped, 266 or more passengers showing up is our event of interest, which is the complement of 265 or fewer showing up.

Calculate Binomial Probability

Use the binomial distribution formula to find the probability that more than 265 passengers show up (which is the same as 266 to 275 passengers). This involves calculating the complementary cumulative binomial probability: \[ P(X \geq 266) = 1 - P(X \leq 265) \] where \( X \) is the number of passengers who show up. Use a binomial cumulative distribution function to find \( P(X \leq 265) \).

Use a Calculator or Software

Using a calculator or statistical software, such as R or Python with a binomial cumulative distribution function, input \( n = 275 \), \( p = 0.95 \), and find \( P(X \leq 265) \). Subtract that result from 1 to find \( P(X > 265) \).

Interpret the Result

After computing \( P(X > 265) \), interpret this as the probability that more than 265 passengers show up, hence someone will get bumped. This value represents the risk of overbooking leading to passengers not getting seats.

Key concepts

Probability Basics

Probability is a way to quantify the likelihood of a particular event happening. It is expressed as a number between 0 and 1. A probability of 0 means the event cannot happen, while a probability of 1 means the event is certain to occur.
To understand probability, imagine we are talking about flipping a coin. The probability of getting heads is 0.5 because there are two possible outcomes (heads or tails), and each is equally likely.
In the context of the airline example, we're examining the probability of a certain number of passengers showing up for a flight based on historical data.

• The binomial distribution is used here to model such probability questions.
• This specific situation involves the probability of 0.05 that a passenger will not show up.

Thus, the probability helps us predict the likelihood of an event, like not having enough seats if more passengers than expected show up.

Understanding Overbooking

"Overbooking" is a common practice in the airline industry. It involves selling more tickets than the actual number of available seats on a flight. Airlines do this based on the assumption that some percentage of passengers will not show up.
This strategy allows airlines to maximize seat occupancy, ensuring the flight is as full as possible.
However, overbooking comes with risks:

• If more passengers show up than there are seats, some travelers will be "bumped" from the flight.
• The airline must compensate bumped passengers, leading to added costs.

The goal of overbooking is to account for no-shows while keeping the likelihood of bumping passengers to a minimum. Calculating this probability allows airlines to create strategies to manage the risk.

Complementary Events Explained

Complementary events is a concept in probability that refers to the idea of two events being mutually exclusive—meaning, they cannot both occur at the same time.
The probability of an event happening and its complement (the event not happening) adds up to 1.

• For the airline, the event of interest is too many passengers showing up.
• Thus, the complementary event is 265 or fewer passengers showing up.

To calculate the probability of the airline needing more seats than available, it's easier to first find the probability of the complementary event (265 or fewer arrivals).
Then, subtract this probability from 1 to find the probability of the original event (more than 265 passengers showing up).
This approach simplifies calculations by focusing on frequencies that are typically easier to find or estimate.