Another user posted this question about elevator occupants, which made me curious about a harder question.
In a $t$-story building (with no basement), $n$ people get on an elevator on the first floor. Each person will uniformly at random independently wish to go to one of the higher floors and will press the corresponding button for the floor assuming it hasn't already been pressed.
What is the probability that three consecutive floors will be visited at some point during the elevator's trip carrying these passengers?
Note, I am not asking for the probability that there are only three floors visited and they all happen to be consecutive, but rather, that among the floors visited, there is a subset of them of size three that are adjacent.
My initial thoughts on the problem is that we might want to approach via "bad words" and strings, letting "+" represent that the floor was visited and "0" represent that the floor was not, we ask for the probability that the sequence of visited or not contains no substring "+++", but this doesn't account for the fact that multiple people might choose to go to the same floor, etc...
An easy approach would be simply to run simulations, but that is uninteresting so I ask if anyone has an idea on a pen+paper approach.
If someone wants specific numbers to work with, try with $n=t=10$, as that was how I mistakenly read the other user's question at first glance.