The Last Seat

Nov 07, 2015

RAY: You're one of a hundred people standing in line to get onto an airplane that has 100 seats. There's a seat for every person who's in line, and each of you has a boarding pass for your assigned a seat. The first person to walk onto the plane drops his boarding pass and, instead of picking it up, decides, "I'm just going to sit anyplace." He takes a seat at random.

Now, every other passenger will take either his assigned seat or, if that seat is taken, that passenger will take any seat at random.

TOM: I've been on that flight!

RAY: You are the last passenger to walk onto the plane. Obviously, there's going to be one seat left, because everyone else is sitting in his correct seat, or not.

The question is: What are the chances that you get to sit in your assigned seat? I'm going make this multiple choice.

A: 1 out of 2.
B: 1 out of 10.
C: 2 out of 50.
D: 1 out of 100
E: Zero.
 

Answer: 

RAY: Our guy with the lost boarding pass, he could by dumb luck wind up in his assigned seat, right? In which case each subsequent passenger including me, if I'm the last person, will get the right seat.

But what's more likely is that he takes someone else's seat.

Now, here's the counterintuitive part. There are only two seats that count, my seat and his seat. All the other seats don't make a difference. Whoever gets on the plane subsequent to him, those people are going to take some seat or another.

Nothing matters until either my seat gets taken or his seat gets taken. If my seat gets taken by some displaced passenger, then I have zero chance of getting my seat. If his seat gets taken by some displaced passenger then every other passenger who walks onto the plane including me, will have his assigned seat.

And how often does that happen? One time out of two. Half the time my seat's going to get taken by some displaced passenger, and half the time his seat is going to get taken by a displaced passenger. So your chances are A) one out of two. Do we have a winner?

TOM: We do. The winner this week is Eliot Gazanski from Ann Arbor, Michigan. Congratulations!

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