## Tuesday, August 03, 2010

### Random Seating is Awful

Image via Wikipedia
People are waiting in line to board a 100-seat airplane. Steve is the first person in the line. He gets on the plane but suddenly can't remember what his seat number is, so he picks a seat at random. After that, each person who gets on the plane sits in their assigned seat if it's available, otherwise they will choose an open seat at random to sit in.

The flight is full and you are last in line. What is the probability that you get to sit in your assigned seat?

1. What, no answers to this one? Did I finally stump everyone?

2. 1/100!
Although I'm excited about the answer, that is supposed to be factorial.

3. 1 in 2, or 50/50?
Either he gets to sit in his assigned seat or he doesn't??

I think its 1/100 for the following reason - if the first person on the plane chooses his own seat (1/100 chance) then every subsequent passenger will choose their own seat, and therefore the last person will get their own seat. Equally, if the first person sits in someone elses seat then each subsequent person has a reducing chance of getting their assigned seat until the last person has only one seat left to occupy, and 99/100 chance that their ticketed seat has gone.

5. HAY! Look at the ticket, it tells you!

6. There are a couple of good explanations out there (see: http://spontaneoussymmetry.wordpress.com/category/puzzles/ or http://discuss.fogcreek.com/techInterview/default.asp?cmd=show&ixPost=468&ixReplies=9), but the answer is 1/2.

If you want to work your way through it, start with just two people on the plane, then three. Look for the pattern.

Another way is to think about what's really happening (see http://flickeringtubelight.net/Tidbits/probablyIrrelevant.php and scroll down to the bottom).

7. 1/100

8. 1/2
as the only possible cases at last are either he will get the seat or not!!!!!!
as simple as that

Leave your answer or, if you want to post a question of your own, send me an e-mail. Look in the about section to find my e-mail address. If it's new, I'll post it soon.

Please don't leave spam or 'Awesome blog, come visit mine' messages. I'll delete them soon after.

Enter your Email and join hundreds of others who get their Question of the Day sent right to their mailbox