One Seat Left. Is It Yours?

Oct 04, 2004

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: Because you are such a kind, generous, and accommodating person, 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?

TOM: A snowball's chance in hell!

RAY: 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.

Think you know?
RAY: I made it multiple-choice. And the options were A) one out of two, B) one out of ten, C) two out of 50, which happens to be one out 25, D) one out of 100, and E, zero.

Now our guy with the lost boarding pass, he could by dumb luck wind up in his assigned seat, right?

TOM: He could.

RAY: 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. It's much more likely.

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.

Are you with me?

TOM: I'm with you.

RAY: 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.

TOM: So your chances are A) one out of two.

RAY: 50/50. Hard to believe! Do we have a winner?

TOM: We do. The winner this week is Eliot Gazanski from Ann Arbor, Michigan. And for having his answer selected at random from the thousands of correct answers that we got, Eliot will win a $26 gift certificate to the Shameless Commerce Division at where he can get among other things a distressed Car Talk T-shirt. And trust me you will look distressed when you're wearing it, Eliot.

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