Puzzle – Take your seats
April 7, 2010 – 1:22 amLet’s say there are 100 people, including you, waiting to board a fully booked flight, each with an assigned seat. The first person to board doesn’t pay any attention to his assigned seat, and sits somewhere at random. Each of the subsequent people will sit in their assigned seat if they can, but if they find someone sitting there will pick another seat at random. You’re the last to board. What’s the probability you’ll get to sit in your assigned seat?
After starting toying with conditional probability, I realized that there are only two seats that really matter as each passenger boards: passenger #1′s seat (let’s call that seat #1), and my seat (let’s call that seat #100). As soon as someone sits in seat #1, it’s all over – I’ll definitely get to sit in my seat. Also, as soon as someone sits on seat #100, I’m done – there’s no way for me to sit in my seat. If any passenger chooses seat #1 or seat #100, we can essentially ignore all subsequent events.
Here’s the key: until either seat #1 or seat #100 is taken, each passenger that boards has the same probability of choosing seat #1 as they do seat #100.
That’s clearly the case for the first guy – he has a 1/100 chance of choosing seat #1, and a 1/100 chance of choosing seat #100. Now let’s take passenger #2. Regardless of what #1 has done, there’s be an equal probability he’ll choose seat #1 or seat #100. That probability is either going to be zero if he’s able to sit in his own seat, or if he can’t because passenger #1 has taken it, he’s equally likely to choose seat #1 or seat #100 (a 1/99 chance in either case). Let’s say we get to passenger #99. The only way the issue won’t have been already decided is if the choices he has left are seat #1 and seat #100. In that case, there’s still an equal probability (50/50 this time) he’ll choose seat #1 or choose seat #100.
So you can see from the above that the chance you get to sit on your own seat is 50/50, because as each passenger boarded there was an equal chance of the question being answered in either of the two possibilities.
Anybody have a different way to get the answer – or can get the answer by looking at the sum of conditional probabilities?
2 Responses to “Puzzle – Take your seats”
What a great problem! Where did you find it?
By Peter on Apr 9, 2010
I seem to recall a wise but somewhat scruffy dude told it to me…
By nolfonzo on Apr 9, 2010