March 2018
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My March diary included the following brainteaser.
You are last in a line of 100 people waiting to watch a play in a theater with 100 seats. Everyone has an assigned seat, but one of the people in front of you is a Free Spirit who will ignore his ticket and choose a seat at random (which might of course be his own).
The other theatergoers will all obediently follow instructions, except that if one of them finds someone else sitting in his seat he will also take a seat at random.
What are the odds that when everyone is seated you will be in your assigned seat?
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• Solution
The odds that you will end up in your assigned seat are fifty percent.
Every theatergoer in front of the Free Spirit will take his assigned seat, so they can be ignored. The Free Spirit will:
- sit in your seat, or
- sit in his own seat, or
- sit in someone else's seat.
If A then obviously you don't get your assigned seat.
If B then you do get your assigned seat, because all of the remaining theatergoers will follow instructions, and all of their seats are free.
If C, then the displaced theatergoer effectively becomes the new Free Spirit, and the process repeats. The new Free Spirit will either:
- sit in your seat, or
- sit in the seat of the original Free Spirit, or
- sit in someone else's seat.
The minute anyone sits in your seat (event A) you lose, while the minute anyone sits in the original Free Spirit's seat (event B) you win.
At every step in the process the odds of A and B are equal, so by symmetry you must have a fifty percent chance of getting the correct seat.