A Puzzle For You All

[TheNewNo2]

I personally proved it by induction on the number of seats. Then I realised you could go more general and instead of proving that the last one had a probability 1/2, you can instead prove that the jth person has probability (n-j+1)/(n-j+2) of getting their own seat (by induction on j), and then the last person is just a special case of that.


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I also rather liked the one from two weeks ago:

There is a long straight single lane road, with all traffic going in the same direction. If each car travels at a unique speed, and overtaking is not possible, how many groups of cars will the traffic separate into over time?

 

[Trog]

I think the chance of you getting your seat is 99%, as for each person getting on the train the chance that Mr Random is sitting in their seat is 1 in 100, So if you are the unlucky one everyone else will have their own seat and they and their seats cancel out. If Mr Random sits in someone earlier in the queues seat he will then start a cascade of one person being out of place in a seat that is both not theirs and unclaimed, which will end if any displaced person sits in the seat allocated to Mr Random, in which case you get your seat. As the number of seats left decreases the chances of this happening gets greater and greater.

Second thoughts there is also the risk of one of the people in the cascade taking your seat, I suppose on average the cascade will start half way through loading and be cancelled out say three quarters of the way to the end as the chance of cancellation increases. So that would make three eighths of the passengers floaters each with a 1 in100 chance of taking your seat. Leaving you with a 5/8 chance of getting your seat say 62% ish.

 

[DaleCooper]

being British, are too polite to do more than tut if their seat is taken

The quoted wording of the question does allow another possible solution. If your seat is occupied by anyone other than the first, very rude and un-British, passenger you just ask the person to move which, being British and polite, they will do and they will move to the remaining unoccupied seat. Thus the probability of getting your own seat is 0.99.

This what Martin Gardner, writer and Scientific American columnist, called a "Quibble".

 

[backontrack]

But that depends whether or not you are British and polite. If you are, then you won't ask that person - you'll just tut. Therefore, we need to know the probability of you being British and polite - but I'm not sure we have enough information...

 

[DaleCooper]

But that depends whether or not you are British and polite. If you are, then you won't ask that person - you'll just tut. Therefore, we need to know the probability of you being British and polite - but I'm not sure we have enough information...

Good point, so a solution isn't possible.