Generalize airplane problem

Question

The well-known airplane problem is can be found here. I restated the problem below.

$100$ women board a plane with $100$ seats. Each of them has a seat assigned in advance. For some reason the first woman who gets in takes a seat randomly. Then the second passenger takes her allocated seat if it is not occupied (by the first passenger), or picks a seat randomly if her own seat is occupied. The third passenger takes her own if not occupied by one of the first two ladies, or a random seat if it is.. and so on.

I would like to generalize the problem to $m$ passengers and $n$ misplaced passengers with $n\leq m.$ The problem above is when $m=100$ and $n=1$.

More precisely,

Given $m$ passengers preparing to board a plane with $m$ seats, if there are $n$ passengers which may not follow their seats, then what is the probability that the last passenger can get their seat?

For the new question, I have no idea how to start at all.

Have you done some calculations with small $m$? Or considered the case $n = m-1$? — hardmath, Dec 31 '19 at 14:20
Does this answer your question? Probability of Sitting in Correct Seat — , Aug 20 '21 at 05:39

Ross Millikan · Accepted Answer · 2020-01-02T19:21:38.273

2

The same argument works. We only care about the order the seats for the misplaced passengers and the last passenger are filled. All permutations are equally likely. The chance the last passenger's seat is the last of these is $\frac 1{n+1}$.

edited Jan 02 '20 at 19:21

answered Dec 31 '19 at 14:45

Ross Millikan

374,822

When $n=1,$ your suggestion leads to probability $1$, where the true probability is $\frac{1}{2}.$ Am I misunderstanding something here? – Idonknow Jan 02 '20 at 12:10
You are correct. I forgot to add in the last passenger. – Ross Millikan Jan 02 '20 at 14:49

Generalize airplane problem

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