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Hamlet is preparing a play to find out the truth about his father's death. The theatre has $2015$ numbered seats and can contain all the members of King Claudius's court. He gives to each member a ticket with the number of their seat.
King Claudius (the first to sit) doesn't look at his ticket and chooses at random a seat. All the other guests arrive one by one and if their seat is available they sit according to the ticket, if it isn't they sit at random.
Ophelia is the last to enter the theatre and she occupies the only empty spot.

What's the probability that she will sit in the seat assigned to her by her ticket?

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  • $\begingroup$ I hope it's not a duplicate; if it is, let me know and I'll delete this. $\endgroup$ – leoll2 Apr 15 '15 at 19:10
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    $\begingroup$ But the real question is where do Rosencrantz and Gildenstern sit??? $\endgroup$ – Ian MacDonald Apr 15 '15 at 19:37
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    $\begingroup$ Doesn't matter, @IanMacDonald, they're dead. $\endgroup$ – Duncan Apr 15 '15 at 22:53
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    $\begingroup$ 50/50 - she'll either sit on it or she wont ;P $\endgroup$ – G.Rassovsky Apr 16 '15 at 9:05
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    $\begingroup$ @JoeZ. surely that's why Ophelia takes her seat last? Because she's (drumroll) ... late (ba-dum tish) $\endgroup$ – Joe Apr 17 '15 at 12:21
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If the king sits in his own seat, then each guest will sit in their own seat and Ophelia will always sit in her own seat. This occurs with probability $1/2015$.

If the king sits in Ophelia's seat, then each guest will sit in their own seat and Ophelia will end up sitting in the king's seat. This also occurs with probability $1/2015$.

If the king sits in any other seat, then as soon as that guest arrives, he will choose another seat at random.

  • If the guest chooses the king's seat, then Ophelia will end up sitting in her own seat.
  • If the guest chooses Ophelia's seat, then Ophelia will end up sitting in the king's seat.
  • If the guest chooses another guest's seat, then as soon as the guest who was assigned that seat arrives, he will also have to choose one at random, and the cycle will continue until one of the above two scenarios happens.

Since the probability is equal that each guest will choose either the king's seat or Ophelia's seat, the eventual overall probability of each event is $1/2$. The above scenario happens with probability $2013/2015$.

So the grand total probability that Ophelia ends up in her own seat is $1/2015 + 0 + 1/2 \times 2013/2015 = 1/2$.

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    $\begingroup$ Brilliant! I knew there'd be a quick and simple way to see the answer to this one. $\endgroup$ – Rand al'Thor Apr 15 '15 at 19:25
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Here's a simpler way to reach the answer:

If anyone sits in the King's seat before Ophelia's seat is occupied, then the rest of the guests will all sit in their assigned seats, and Ophelia will be in her seat.

If anyone sits in the Ophelia's seat before the King's seat is occupied, then Ophelia obviously cannot be in her seat.

But none of the guests know which seat is the King's and which is Ophelia's! Therefore the two scenarios are equally likely, and Ophelia has a probability of 1/2 of ending up in her assigned seat.

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Yet another way to the answer;

There is exactly one guest(including the king) sitting in a seat that we do not know belongs to a guest that has already arrived.

This means that when a guest arrives he will find his seat taken with probability 1/(Number of free seats+1). This continues until Ophelia arrives, so Ophelia has probability 1/(1+1) of finding her seat taken.

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My guess and an attempt at a proof:

P(x) = 1 - [1/2015 + ΣP(1/(2015-x))] Where x is the seat # of the person who's seat was taken. Initially there is a 1 in 2015 chance the king will take Ophelia's seat. Then there is a 1/(2015-X) chance that the person who's seat the king took will take Ophelia's seat...Continue the process for all the peoples who's seats were taken, add all the probabilities up and you should get the chance her seat is taken. One minus that probability is the chance the seat is empty. The end result should be 1/2 as stated by Joe Z.

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    $\begingroup$ I'm guessing this will eventually evaluate to 1/2. $\endgroup$ – Joe Z. Apr 15 '15 at 19:26
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    $\begingroup$ The king can take his own seat! It's rare, but he can! $\endgroup$ – leoll2 Apr 15 '15 at 19:40
  • $\begingroup$ I think I fixed it. :) $\endgroup$ – Mark N Apr 15 '15 at 19:42
  • $\begingroup$ I still can read this sentence in your post. "Assume The king will always take 1 seat that wasn't his" The king can assume any place! $\endgroup$ – leoll2 Apr 15 '15 at 19:56

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