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n passengers board a plane with n seats. Each passenger has an assigned seat. The first passenger forgets and takes a seat at random. Every subsequent passenger sits in their assigned seat unless it is already taken, at which point they take a random seat. What is the probability (in terms of n) that:

Every passenger sits in the correct seat?

At least one passenger sits in the correct seat?

Exactly one passenger sits in the correct seat?

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Every passenger sits in the correct seat?

This only happens if the first passenger choses his own seat by accident, the probability of which is $1/n$.

At least one passenger sits in the correct seat?

This is the complementary event of noone being in their correct seat. For that to happen passenger 1 must pick the seat of passenger 2, passsenger 2 must pick the seat of passenger 3, etc. The probability of that is $1/n\cdot1/(n-1)\cdot\ldots\cdot1/1=1/(n!)$. The complementary event has probability $1-1/(n!)$

Exactly one passenger sits in the correct seat?

This happens if every passenger picks the seat of the next passenger except for one, which picks the seat of the second next passenger. The first passenger cannot be in his correct seats.

Probability of the $i+1$th passenger being in their correct seat: $$1/n\cdot 1/(n-1)\cdot ...\cdot 1/(n-i+1)\cdot 1/(n-i-1)\cdot ...\cdot 1/1=(n-i)/n!$$

It can be anyone of passenger $1$ to $n-1$, yielding a total probability of \begin{align*} \sum_{i=1}^{n-1} (n-i)/(n!)&=1/(n!)\sum_{i=1}^{n-1} n-i=1/(n!)\cdot(n(n-1)-1/2\cdot(n-1)n)\\ &=1/2\cdot 1/(n!)\cdot (n-1)n=1/2\cdot 1/((n-2)!) \end{align*}

An extra question for you: What is the probability of exactly one passenger being in the wrong seat?

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  • $\begingroup$ I agree with your derivation but I'm unclear why the last passenger cannot sit in their correct seat. If the 1st sits in 2nd's seat and 2nd sits in 3rd's seat and so on up to second to last passenger. The second to last passenger can sit in the last seat or the first seat. If the second to last passenger sits in the seat that was for the first passenger, then the last passenger must sit in their own seat. Please correct me if their is a fault in my logic. $\endgroup$ – Inquirer Sep 28 '18 at 16:15
  • $\begingroup$ You are right. I forgot about the first passenger. Good thing you spotted it. $\endgroup$ – Asdf Sep 28 '18 at 18:21

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