“A man walks into a men’s room…” No, it’s not the beginning of a joke, it’s the beginning of a scholarly paper presented in Lecture Notes in Computer Science, 2010, Volume 6099/2010, 284-295.
To continue :
“A man walks into a men’s room and observes n empty urinals. Which urinal should he pick so as to maximize his chances of maintaining privacy, i.e., minimize the chance that someone will occupy a urinal beside him?”
Professor Evangelos Kranakis (School of Computer Science, Carleton University, Ottawa, Ontario, Canada.) and professor Danny Krizanc (Department of Mathematics and Computer Science, Wesleyan University, USA.) undertook a complex mathematical study of such a scenario and scrupulously analysed its manifold possibilities – aiming, of course, at a clarification.
“It turns out the answer depends on many things but in particular on how one models the behavior of the men who enter later.”
– say the authors.
“Our main conclusion is that when faced with the decision of what urinal to choose upon entering the men’s room, in order to maximize your privacy, you should probably choose the one furthest from the door if it is available and the one next to it is unoccupied. For a vast majority of the (what we consider) natural behaviors that men choosing urinals might follow, this choice is optimal (in the sense defined above).”
And, going beyond this observation, the authors urge additional consideration ;
“…we feel that this problem leads to many interesting variations that are worthy of investigating further and we encourage everyone to do more of their thinking while using public restrooms.”
The paper can be perused in full here
Authors’ note for female readers :
“We hope that by focusing on the men’s room version of the problem we are not leaving out our female readers. We chose this version since (1) we understand (not from personal experience) that most women’s rooms contain only water closets which already provide some amount of privacy beyond that afforded by urinals and (2) we are more familiar with the behavior of men in a public restroom situation which we shall see is an important aspect of our study. It has been pointed out to us that women may also use urinals  and should this become a more prevalent behavior our results may directly interest our female readers.”