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Have you ever sat in a doctor's[br]office for hours

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despite having an appointment[br]at a specific time?

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Has a hotel turned down[br]your reservation because it's full?

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Or have you been bumped off a flight[br]that you paid for?

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These are all symptoms of overbooking,

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a practice where businesses[br]and institutions

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sell or book more [br]than their full capacity.

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While often infuriating for the customer,

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overbooking happens because[br]it increases profits

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while also letting businesses[br]optimize their resources.

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They know that not everyone[br]will show up to their appointments,

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reservations,

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and flights,

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so they make more available[br]than they actually have to offer.

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Airlines are the classical example,[br]partially because it happens so often.

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About 50,000 people get bumped[br]off their flights each year.

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That figure comes at little surprise[br]to the airlines themselves,

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which use statistics to determine[br]exactly how many tickets to sell.

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It's a delicate operation.

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Sell too few, and they're wasting seats.

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Sell too many, and they pay penalties -

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money, free flights, hotel stays,[br]and annoyed customers.

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So here's a simplified version[br]of how their calculations work.

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Airlines have collected years worth[br]of information

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about who does and doesn't show up[br]for certain flights.

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They know, for example,[br]that on a particular route,

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the probability that each individual[br]customer will show up on time is 90%.

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For the sake of simplicity,

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we'll assume that every customer[br]is traveling individually

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rather than as families or groups.

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Then, if there are 180 seats on the plane[br]and they sell 180 tickets,

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the most likely result is that 162[br]passengers will board.

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But, of course, you could also[br]end up with more passengers,

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or fewer.

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The probability for each value[br]is given by what's called

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a binomial distribution,

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which peaks at the most likely outcome.

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Now let's look at the revenue.

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The airline makes money from each[br]ticket buyer

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and loses money for each person[br]who gets bumped.

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Let's say a ticket costs $250[br]and isn't exchangeable for a later flight.

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And the cost of bumping [br]a passenger is $800.

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These numbers are just for the sake[br]of example.

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Actual amounts vary considerably.

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So here, if you don't sell[br]any extra tickets, you make $45,000.

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If you sell 15 extras[br]and at least 15 people are no shows,

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you make $48,750.

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That's the best case.

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In the worst case, everyone shows up.

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15 unlucky passengers get bumped,[br]and the revenue will only be $36,750,

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even less than if you only sold 180[br]tickets in the first place.

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But what matters isn't just how[br]good or bad a scenario is financially,

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but how likely it is to happen.

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So how likely is each scenario?

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We can find out by using[br]the binomial distribution.

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In this example, the probability[br]of exactly 195 passengers boarding

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is almost 0%.

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The probability of exactly 184 passengers[br]boarding is 1.11%, and so on.

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Multiply these probabilities[br]by the revenue for each case,

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add them all up,

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and subtract the sum from the earnings[br]by 195 sold tickets,

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and you get the expected revenue[br]for selling 195 tickets.

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By repeating this calculation[br]for various numbers of extra tickets,

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the airline can find the one likely[br]to yield the highest revenue.

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In this example, that's 198 tickets,

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from which the airline will probably[br]make $48,774,

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almost 4,000 more than without[br]overbooking.

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And that's just for one flight.

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Multiply that by a million flights[br]per airline per year,

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and overbooking adds up fast.

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Of course, the actual calculation[br]is much more complicated.

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Airlines apply many factors[br]to create even more accurate models.

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But should they?

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Some argue that overbooking is unethical.

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You're charging two people [br]for the same resource.

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Of course, if you're 100% sure [br]someone won't show up,

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it's fine to sell their seat.

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But what if you're only 95% sure?

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75%?

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Is there a number that separates being[br]unethical from being practical?