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Welcome to our last lecture of this module on Discrete Random Variables and

Â Probability Distribution.

Â In this last lecture I'm going to talk about a very important application of

Â the binomial distribution and probabilistic thinking in general.

Â Airlines, airlines overbook their flights.

Â Which means, let's say, there is a flight from Zurich to London that has 200 seats,

Â airlines may actually sell 210 tickets or 220 tickets.

Â Why are they doing it?

Â What are the benefits and the costs of doing this?

Â Airlines face a big problem, last minute cancellations or total no-shows.

Â People may cancel a flight, just a couple hours before it's supposed to take off,

Â and they still may get a substantial refund.

Â At this point, the aircraft may leave with empty seats.

Â That's bad for the airlines, because there's lost revenue.

Â They could have sold a seat, they could have made more money.

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The case from one second to a next, it was available for

Â sale and it's gone, and the airline cannot sell it anymore.

Â It went bad, it went moldy, sort of in an instance and that's very bad.

Â So, the airlines wants to get extra revenues and

Â one way is to do airline over booking, that's the benefit.

Â The cost on the other hand is now when too many people show up.

Â You have only 200 seats, you sold 210 tickets, suddenly 205 people show up.

Â Now you have to buy some people out of traveling, either by giving them vouchers.

Â Or in the worst case, if you don't have enough volunteers who take vouchers,

Â you have to kick off people.

Â That cost you money directly, you may really annoy your customers, and

Â they say, I never fly with that airline again.

Â So you may have serious loss in customer goodwill, and

Â that's difficult to put Swiss Franc or Dollar figure on that.

Â And in general, you have customer dissatisfaction.

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This independence of course is not quite correct.

Â Whenever you have groups travelling, the family of four, parents with their

Â kids maybe all go to a vacation trip, visiting relatives or nobody goes.

Â So clearly there is some dependents or

Â some business people either all go to visit the customer, or nobody goes.

Â Here, we simplify from this and

Â say all the people make the decision, independently with a probability p.

Â And voila, we have a binomial setup, and we can use the binomial distribution.

Â I put together a large spreadsheet for you on airline overbookings.

Â So let's have a look at this spreadsheet.

Â Here but for yellow background,

Â I provided some parameters, that we just take as given.

Â We have an airliner with 200 seats.

Â The price of a ticket is $300.

Â The bump cost is $500, which means in addition to reimbursing

Â an overbooked passenger who cannot fly with us, the $300 for

Â the price, there is also a $500 voucher or hotel cost, or you name it.

Â And we assume a probability of no shows of 0.07.

Â I've found this type of number in some reports on airline overbooking.

Â Here now is the number of tickets sold.

Â Let's say we sell 200 tickets, now look at this.

Â Under the assumption that 7% don't show up, we assume 14 people won't show up.

Â So on average only 186 people show up.

Â So on average, our airliner there is not full.

Â We don't like this, and we see here that revenues would be in this case $55,800.

Â Now let's say we sell 205 tickets.

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Now the expected value of people showing up is

Â still nowhere close to 200, tt's now 190.65.

Â Notice the expected revenue went up.

Â How did I calculate those?

Â Now let's go deep into the spreadsheet here.

Â If we sell 205 tickets,

Â then in the worst case 205 people show up.

Â My revenue, however, is only 60,000.

Â Why, because only 200 people will actually leave with us, 200 times 300.

Â Then I have the over cost of 2,500.

Â That means five people get over cost of 500, five people get 500 each,

Â 2500, in addition to the 300 that I don't get,

Â makes revenue minus cost 57,500.

Â Now how likely is it that that happens?

Â Here, look at this probability,

Â it's a binormal distribution of having exactly 200 people show up.

Â If I sell 205 people show up, if I sell 205

Â tickets the probability of 93% showing up, that's essentially 0.

Â This worst case scenario just doesn't happen.

Â Let's look at now this case, 202 people show up.

Â Again the revenue's the same, 60,000, I can only take 60,000,

Â make only that because I can take only 200 with me,

Â 2 people I have to bump, that's 500 in overage cost.

Â So I make net only 59,000, not 60,000, only 59,000.

Â What's the probability of that happening?

Â Still very tiny, and again I use a binomial distribution.

Â 202 people show up, if I sell 205 tickets, but for

Â probability of showing up of 93%, that's one minus the 7%.

Â And so, now I have all the possible revenue minus cost.

Â Here I have all the possible probabilities, and

Â then I can calculate the expected value, which I do here in the column J.

Â I add up all these numbers.

Â Now, I can tell you the truth,

Â I don't actually do this for all the possible numbers.

Â Why, because at some point they're all zero, the probabilities are so tiny.

Â And here, I get the round number to two digits, 57,193.

Â Now, let's play with this, let's say we sell five more tickets, 210.

Â What happens to revenue minus cost?

Â It's still goes up.

Â Let's try 214, it still goes up!

Â Now let's go crazy, 230, not a good idea, it goes way down.

Â Why, because, suddenly it's very likely that we have to bump many people.

Â For example, if we sell now 230 tickets,

Â the probability of say 214 showing up is 10%.

Â That's rather high, so clearly it's not a good idea to sell 230 tickets.

Â Now how can you find the optimal number?

Â Here with some trial and error, you can play around with this number, and

Â it's around 214, 215 is the optimal number.

Â Now for those of you who know a little bit more Excel, if you have access to

Â the Excel server, I have hooked up an Excel server to this Excel sheet,

Â and with this server we can actually find the optimal solution here.

Â However that requires some optimization, which is not part of this class.

Â But for any people who have played around with the server before,

Â you may find that interesting.

Â Now let me summarize what we have seen here in the slides.

Â So, and move back to the slides.

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And in the spreadsheet we saw how we can use probabilities to sort of find

Â the optimal, the maximized expected payoff to the airline.

Â Now clearly some assumptions went into our calculation, but

Â I think there's one very robust takeaway.

Â It is not optimal for airlines to only sell as many tickets as they have seats.

Â It really is necessary for them to overbook their airlines, their flights.

Â To summarize, we have now seen in our last lecture in this module,

Â an application of the binomial distribution, namely airline overbooking.

Â This is just one example of this large area called Revenue Management.

Â And this concludes the module on Discrete Random Variables and

Â Discrete Probability Distributions.

Â In our next module now, we will take a look at continuous distributions.

Â We will see the famous bell curve and

Â the normal distribution with some cool applications.

Â So please stay with us and come back for the next module of

Â An Intuitive Introduction to Probability, Decision Making in an Uncertain World.

Â Thank you.

Â