## Bumpy Flight

The holidays are rolling around again, and the time is fast approaching to book your flight home.  But this year you're ahead of the curve.  You've combed all the travel sites, checked your fare alerts, sold some plasma, and pulled the trigger on that window seat you've had your eye on since mid-August.  Well done.  It must be nice to have one fewer empty box on the ol' checklist.  Maybe.

But here's the dirty little not-so-secret of the airline industry: somebody else may also have bought your beautiful window seat.  Which is weird, because you probably assumed that making a reservation somehow, you knowreserved something.

You'd think that, but you'd be wrong.  The breathtaking slew of industry-wide financial implosions notwithstanding, airlines aren't terrible at math.  They know that not everyone who books a ticket on a flight will actually show up for that flight, and "no-shows" cost money.  To combat this loss, the airline will sell more tickets than there are seats, hoping that exactly the right number of people will show up come boarding time.  Their hoping is very scientific, of course, but that's the gist of it.  So let's see how this might play out.

The market for airline tickets is insanely complex, as is passenger behavior, so we're going to have to make some simplifying assumptions to even begin to get a handle on things.  For starters, let's look at a single flight on a Mathalicious Air (motto: "We kind of like to fly, and it shows from time to time") Boeing 737, which might reasonably have 150 seats.  No-show rates vary for a lot of reasons, but 10% is a reasonable hypothetical average, so let's go with that.  What would happen if we were to sell exactly 150 tickets for our most popular flight, the 8 a.m. from Charlottesville to Charlotte?

Well, on average, about 135 people would actually show up at the gate, which isn't awesome for us, since we could have sold 15 more tickets to fill up the flight while we waited for the no-shows to rebook.  What if, instead, we used our knowledge about no-shows to try and compensate for the missed opportunity?  In other words, if we were to account for the fact that only 90% of ticketed passengers will make the flight, how many seats would we book?  Ninety percent of what number gives us 150 butts in seats?

So we sell 166 tickets, effectively double-booking 16 seats.  On average, just about 150 people actually show up at the gate, and everybody's happy...right?  Not exactly.  On average, 150 people show up, but about half the time we actually end up with more people than we have space for, and those people need to be compensated for getting bumped to a later flight.  That also costs us money.  Then again, we took in more money by booking the extra seats, so the question is, is it worth it?  How much money is the overbooking costing us?  Is it more or less than the additional revenue we're taking in?

That's a bit of a tricky question, because it depends on how many people we end up bumping.  Let's say our tickets cost \$300 apiece, and our standard compensation package to a bumped customer is a generous \$1000.  How much is overbooking going to cost us, on average?

We can model this situation with a binomial distribution, where we have 166 trials and a 90% probability that a trial will result in a person showing up for her flight (and, of course, a 10% probability that it results in a no-show).

From the distribution we can see that the probability of 151 people showing up for the flight --- and thus the probability that we would have to spend \$1000 to bump a single passenger --- is about 9.8%.  The probability we'd have to spend \$2000 to bump two passengers is about 8.7%.  If we look at the weighted average of all the money we'd have to spend on bumps (i.e. the expected value of our payout), it turns out that an average flight with this level of overbooking would cost Mathalicious Air about \$1244.  But remember, we took in an additional \$4800 by double-booking 16 seats.  We're still making out on the deal!  In fact, we could get away with overselling even more seats if we were so inclined.

Of course, we might not be so inclined.  After all, we're already bumping passengers on more than half of our flights, which isn't a super PR move.  That's a lot of disgruntled customers in a market where brand loyalty is essentially absent.  So even though federal regulations don't do a lot to prevent passengers from being inconvenienced, and even though it's financially advantageous for airlines to oversell their flights, there's a fine line between extra profit and angry consumers.  So keep that in mind when you're heading to the gate in a couple weeks.  Maybe, just maybe, you'll get home on time.

Teachers: want to have this conversation with your students?  Check out the materials on our lesson page!