COVER STORY
Suppose, I have an infinite metro line with an infinite number of stations.
Let single car occupancy be $n \cdot k$, and we let up to $n$ people leave and enter at each station. For each person entering the car we draw a destination station to be one of the $k$ next stations uniformly at random. People with destination at current station leave the car.
I want to ensure, that there are at most $n$ people destined at the next station in the car, i.e. I don't want anyone to miss her respective station.
Suppose, I have calculated the probability for this successful outcome for station numbered $m$, and the probability is $p$. I'm pretty sure $p < 1$.
What makes me worry, is if I look at (possibly) independent events "all passengers to station $m$ successfully left", "all passengers to station $m+k$ successfully left", etc. This is in some sense a geometric random variable, and undoubtedly some will "succeed", i.e. not all passengers could leave at some station.
ESSENCE
But the same kind of idea is applied by airlines, when I make an online seat reservation. My weight is never asked (for at least 150+ passengers flights), nor the weight of kids, or other relatives. Nonetheless, I suppose the airline expects the weight of each row, or more critically the blocks of rows to be "equally" distributed throughout the plane.
Since, there are possible passenger arrangements, which assign all the light passengers, let say, to the front and all the heavier passengers to the back, I deduce the probability of successful passenger assignment $p$ is also less than $1$. I.e. after some finite number of assignments some plane should be loaded in some wrong way.
Why airlines do not seem to worry?