This is actually a real-world problem, although I've changed some details for privacy.
I currently live in an old nursing home that used to house 50 people in individual rooms, but for various reasons, only 5 people currently live here. The landlord is trying to get the remaining 5 people to all move onto the same corridor, probably so they can start to remove bathrooms from the rest of the building ready for demolition. It seems intuitive to me that 5 people sharing one bathroom will provide a lesser quality of living (in terms of waiting to use the bathroom) than 50 people sharing 10 bathrooms, which was the original housing situation i signed up for, and I want to use this as an argument not to move if our case goes to court.
It feels very similar to a queuing problem I remember from an exam, and it also feels similar to a situation where (i would guess) a person finds it easier to fit 100 boxes with variable dimensions into a 100x space, compared to someone trying to fit 10 variable boxes into a 10x space. I'm not confident enough that my guess is correct, so I'm looking for help formalizing the problem.
I feel like the problem could be phrased in terms of the probability that a toilet will be available at any given time, but it could also be phrased in a number of other ways, and terms like "box packing" and "optimization" come to mind, although I could do with being pointed in the right direction here, because I have very little experience with those kinds of problem.
I also sound like a bathroom-obsessive, and there are probably other ways to show a material difference either side of the landlord's proposal, but this seems like a widely understandable one.
Any help would be greatly appreciated.
Edit: Here's my current way of working this out, perhaps someone could say whether it seems valid -
I assume that a bathroom "appointment" is 5 mins long, and everyone takes 3 random appointments of 5 mins a day, so the probability of any particular person occupying a particular bathroom appointment = 3/288.
with 50 people, 10 bathrooms, I then find the probability that there aren't enough bathrooms, or, that 11 people randomly choose a particular appointment:
Probability of any 11 particular people picking the same appointment is 3/288 to power of 11 = 1.5668103674140072093002304132879e-22
But there are various different combinations of people that this could be, the number of those combinations is 5049484746454443424140 = 1,491,041,720,931,840,000
Multiply that number by the basic probability at point 3 = 0.00023361796266028298342184840297394
But this can happen on any of the 288 slots, so 288 times the answer at point 5 = 0.067
I then do a similar calculation for the possibility of 5 people sharing one bathroom and I get 0.625, which feels about right considering how many times in an average house I go to use the bathroom and someone's in there.
Is this working out valid?