Another solution to the Airplane Probability Problem

I found this problem in a post by Brett Berry, and came up with a couple of solutions. The first one took a fair bit of algebraic wrangling on paper. Later, I thought of a second proof, one requiring no math at all.

There are a few steps which can help you tackle problems like this. First, come up with some terminology. This helps you describe the problem, and reason about it further in a convenient way.

1. Let’s call the first passenger, the one who has lost his ticket, Frank.

2. Let’s say that you win if you end up sitting in your assigned seat, otherwise you lose.

3. Let’s say that a mismatch exists if the assigned seat of a passenger who has yet to board is already occupied.

A second step to solving a problem like this is to look for some insights that may lead to a solution.

Insight 1. Throughout the boarding process, there is at most one mismatch that exists at any particular time.

To see why this is true, note that if Frank happens to choose his own seat by chance, then there will be no mismatches at all. Otherwise, there is a single mismatch until the passenger who is supposed to sit in the seat Frank has claimed (let’s call her Alice) chooses another seat to sit in. This creates a new mismatch (with that other seat’s owner, who is still waiting to board); but as Alice is now seated, ‘her’ mismatch no longer exists.

Insight 2. If anyone (other than you) ever sits in Frank’s seat, you’ll win.

If someone sits in Frank’s seat, there will be no more mismatches, since a mismatch is only created if someone whose seat is occupied claims the seat of someone still waiting to board.

Insight 3. This is kind of obvious, but if anyone (other than you) ever sits in your seat, you’ll lose.

Now let’s solve the problem!

When Frank boards the plane, he’s effectively rolling a 100-sided die. One of the sides has his seat number on it, and one of them has yours. The other 98 sides have other passenger’s seat numbers on them. If he sits in his seat, you win. If he sits in yours, you lose. Otherwise, he creates a mismatch for some other passenger.

Suppose it’s someone else’s turn to board. She takes her assigned seat, if it’s available. Otherwise, she’s got a seat mismatch, and takes one at random. This is also like a rolling a die, one with between 2 and 99 sides. One side has Frank’s seat number on it, one has yours, and the outcome of this roll is a win, a loss, or another mismatch.

Of all of these rolls, one must produce a win or a loss, since by the time you get to board the plane, there’s only one remaining unoccupied seat (which means someone must have sat in either your seat or Frank’s). With each of the dice, the odds of rolling a ‘win’ is the same as rolling a ‘loss’, regardless of when it occurs. Your odds of sitting in your assigned seat are therefore 50/50.

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