Basic probability and counting methods A somewhat geeky problem has been on my mind the last few days: In my accommodation at Uppsala, there are 12 rooms to a floor. I discovered the other day that another British girl whom I know lives directly beneath the room in which I live. 
What is the probability of this happening? My immediate instinct is to think $\frac{1}{12}$, though I can't help but think that the mathematics is slightly more complicated than this. Is it as simple as $12\times ((\frac{1}{12})^2)$? Is this generally the correct method to work out probabilities of this kind? If so, why? I'm RUBBISH at counting methods.
Thanks :)
 A: The probability that the girl is in the room directly below you depends on a number of factors you have not considered.
How many floors are there in the accommodation?  You first have to consider the probability that you are on the floor directly above her. 
You also have to consider bias in assigning guests to floors and rooms.  Are higher level floors more expensive than lower (and what is the likelihood that you and she pay for what price rooms)?  What is the layout of single and double rooms, etc, likewise are male and female guests evenly distributed or segregated?  Is this layout the same on all floors, alternating, or otherwise dependent of level?

That said:  Let's consider that there are only two floors, and that there is no apparent bias to assigning guests to floors, nor to rooms on floors.
The probability that you are on the second floor, she is on the first, and that you are directly above her is:
$$\begin{align}
\mathsf P(f(U)\!=\!2, f(S)\!=\!1, r(U)\!=\!r(S))
& = \mathsf P(f(U)\!=\!2)
\;\mathsf P(f(U)\!=\!1\mid f(S)\!=\!2)
\;\mathsf P(r(U)\!=\!r(S)\mid f(U)\!=\!2, f(S)\!=\!1)
\\ & = \frac {12}{24}\cdot\frac{12}{23}\cdot\frac{1}{12}
\\ & = \frac{1}{46}
\end{align}$$
If you know that you are on the floor above her then the conditional probability that she is directly below you is $1/12$, whatever room you are in.
And so forth...
It's a little more complex if there are more floors.
