Puzzle – Take your seats

Another problem that has an immediate intuitive answer, but can be tricky if you don’t have that insight.

Question

Let’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 has seat #1 assigned but doesn’t pay any attention to this 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 with seat #100 assigned. What’s the probability you’ll get to sit in your assigned seat?

Answer

OK so if you’re really smart you may be starting to toy with conditional probability.

But an insight that can solve this quickly is that as soon as anyone sits in seat #1, I’ll be guaranteed my seat. Conversely, as soon as anyone sits in seat #100, my seat is eliminated and I won’t get to sit on it.  In other words, there will be one passenger that seals my fate, and all other passengers are irrelevant.

The passenger that sealed my fate had a 50/50 chance of either guaranteeing or eliminating my seat.  That applies for passenger #1, who had a 1/100 chance of choosing his own seat #1, and a 1/100 chance of choosing my seat #100.  If he’s the one that gets to seal my fate, it’s 50/50 which way he’ll do it.  If it gets to passenger #99 without my fate being sealed, #99 will have the pleasure of definitely doing so with only seats #1 and #100 left, which he’ll choose at random as neither is his.

So you can see that the chance you get to sit on your own seat is 50/50, because the passenger that sealed your fate had a 50/50 chance of doing it either way and all other passengers who chose seats other than #1 or #100 are irrelevant.  And your fate will be definitely be sealed somewhere between passenger #1 and #99.

Anybody have a different way to get the answer – or can get the answer by looking at the sum of conditional probabilities?