This might sound like a mistake but I can assure you it isn’t. We live in a world where such operational mistakes, especially in the airline industry, do not happen unless it’s on purpose.
So this is not a mistake but a risk taken by the airline to maximize profit. In more fancy terms, this is a statistically predictable trade-off and it is based on data science (or pure math). This was a decision based on probability and math to potentially gain some extra millions.
In this article, we will get to the bottom of this and learn a lot of valuable concepts around probability, distributions, and expected value (highly important subjects for data scientists, analysts, or anyone interested in analytics).
DS Airlines Selling Extra Tickets
I’d like to call this company DS Airlines. They sell 304 seats for a flight with a capacity of 300 seats. The probability of a passenger showing up for a flight is 95% based on historical flight data.
We will try to answer the following questions:
- What is the probability that this flight is overbooked? It happens if at least 301 passengers show up for the flight.
- What is the expected value of the number of overbooked passengers?
- What is the airline risking when selling more tickets than available seats?
We start by making the assumption that passengers are independent from each other. One passenger’s choice of showing up for the flight has zero impact on other passengers’ choices. Then, we can say that the probability of any passenger showing up is 0.95 (i.e. 95%).
This assumption does not always hold because there might be families or groups flying together. If one person in a family shows up, then other members of the family are much more likely to show up than 95%. However, when we get into such details and conditions, the math becomes a nightmare with complex conditional probabilities. Besides, we may not always know who are flying together (e.g. members of a group can purchase tickets separately).
With the assumption of independent passengers, we can say that the number of passengers who show up for the flight follows binomial distribution.
Binomial Distribution
Binomial distribution is a probability model used for counting successes in a series of repeated, identical, and independent events. The outcome of these events must be binary (i.e. 1 or 0, show up or do not show up for the flight). Hence, 304 passengers showing up for the flight can be modeled using a binomial distribution of a series of 304 independent events where the probability of an event is 95%.
For a scenario to be qualified for a binomial distribution, it needs to satisfy the following conditions:
- Binary outcome: Success (show up for the flight) or failure (does not show up).
- Independent events: A person showing up for the flight has zero impact on other passengers.
- Same probability of success: It’s 95% for all the passengers.
- Fixed number of events: This is the number of tickets sold for this flight, which is 304.
Binomial distribution helps us answer the following question: If I repeat an experiment multiple times, what is the probability of getting exactly k successful outcomes?
Probability of Overbooking
Our flight will be overbooked if more than 300 passengers show up so we need to calculate the probability of k > 300 successful outcomes.
With binomial distribution, the probability of k number of successful outcomes is:
- n is the total number of events, which is 304
- k is the number of successes
- p is the probability of success event, which is 0.95.
- (1-p) is the probability of failure event (not showing up), which is 1 – 0.95 = 0.05.
This formula relies on the independence of events. In probability theory, if A and B are independent, the probability of both events happening is their individual probabilities multiplied together:
To calculate the probability of k=301 passengers showing up, we multiply 301 0.95 and 5 0.05 together. Then, we multiply it with the combinator at the beginning. It reads as “n choose k”, which counts for how many different ways we select 301 people out of 304.
The following drawing makes it a little easier to grasp the logic behind the formula of the probability with binomial distribution. To simplify things, it illustrates the probability of 4 passengers showing to a flight with 5 tickets sold:
The first row assumes passenger 5 does not show up, the second row assumes passenger 2 does not show up. Since these events are independent and identical, the probability of these events are the same. There are 5 combinations of 1 passenger not showing up (i.e. 5 chooses 4 is 5). Hence, the probability of 4 passengers showing up for the flight is 5 times the probability of 1 passenger not showing up.
So the answer to our first question (What is the probability that this flight is overbooked?) can be calculated as P(X > 300) which is the sum of P(X=301) + P(X=302) + P(X=303) + P(X=304):
The numbers at the beginning come from combinations:
We can select 301 passengers from a group of 304 passengers in 4,636,304 different ways.
So the final result is:
The exact probability that the flight will be overbooked is 0.000139, which translates to roughly 0.014% (or about a 1-in-7,200 chance).
We’ve found the probability that this flight will be overbooked. As we can see in the calculations above, the flight is overbooked if 301, 302, 303, or 304 passengers show up (i.e. 1, 2, 3, or 4 overbooked passengers). This brings us to the second question: What is the expected value of the number of overbooked passengers?
Expected Value
Unlike what its name suggests, expected value is not the outcome we expect to happen on any single trial. The answer to the question “how many passengers do you expect to show up on the flight tomorrow?” is not the meaning of expected value.
Instead, it is the mathematical “long-run” average of a random variable if we repeat the experiment over and over again. So it can be a floating point number such as 302.4 people.
To calculate the expected value, we weigh each potential outcome by the probability of it actually happening. There will be at least one overbooked passengers if 301, 302, 303, or 304 passengers show up for the flight. We can use the probabilities we already calculated for these events:
If 301 passengers show up, there is 1 overbooked passenger. If 302 passengers show up, there are 2 overbooked passengers, and so on. Hence, the expected value of the number of overbooked passengers is 0.000166.
Let’s look at it on a greater scale. If the airline operates this specific flight route 10,000 times, they will have a total of 1.66 overbooked passengers (in all 10,000 flights combined).
This brings us to where we discuss real business value, which is our third question: What is the airline risking when selling more tickets than available seats?
From Pure Math to Real Business
Let’s attach some realistic numbers to our 10,000-flight example, assuming an average ticket price of $200:
By selling 4 extra tickets on every single one of those 10,000 flights, the airline generates 40,000 extra ticket sales. At $200 a pop, that is $8,000,000 in revenue that would have otherwise flown away as empty seats.
Out of those 10,000 flights, we expect to have about 1.66 overbooked passengers. Under consumer protection rules (like the US Department of Transportation), if an airline involuntarily bumps you, they owe you cash compensation of up to 400% of your fare, capped at $2,150. Even taking into account other expenses (e.g. hotel vouchers), the cost to the airline across all 10,000 flights is less than $5,000.
From a strictly financial standpoint, risking $5,000 to secure $8 million in extra revenue is very appealing.
However, life is not pure math. Going viral due to a chaotic gate confrontation can result in thousands of dollars of lost customer loyalty. Depending on the severity of this viral PR, airlines can even lose stock market value. We can all agree that dragging people off planes is terrible for long-term business.
Airlines have a solution to this as well. They run “reverse auctions” via their mobile apps or at the boarding gate. Instead of assigning a computer to kick a random passenger off the flight, they ask: “Who wants to take the next flight for a $500 voucher? $800? $1,500?”
The plane flies at 100% capacity, the revenue stays in the airline’s bank account, and nobody goes viral on social media.