How to model the distribution of a random variable Problem:
Suppose there are two different versions of a piece of software that are distributed among 2*N employees within a company, such that exactly N employees get one version and the rest get the other version. Each employee installs the version on his/her respective computer. The company then hosts a programming competition between N participants and they all need to use the same software. To ensure this, an employee is selected to check the software on each computer until they find exactly N computers that have the same version installed (it doesn't matter which version it is). Upon finishing his task, the employee reports the number of computers that were checked to the project supervisor (K) and the order in which they were checked, but fails to note the version that the participants would be using. The supervisor then needs to check the computers in the same order until he is sure he knows what version of the software to use. 
Let $X$ be the random variable used to model the number of computers the supervisor would need to check in order to be sure of the version that they would have to use. What is the distribution of $X$?
Note: I'm currently working on a piece of software that for the most part doesn't deal with probability, random variables etc. so this is not a field I'm really familiar with. I did some searching, however, but I can't fully commit to reading everything I need to know to tackle problems like the one I'm posting about (it's really the only one I need to deal with in the entire project), so I would appreciate if you guys could help me out.
 A: For the employer to know the order in which the computers were checked, he must have been noting the first computer as version $A$ and the first computer with a different version than $A$ as $B$ and then forgetting what version $A$ and $B$ correspond to. An example of a sample path of him noting:
A, A, B, A, B, B, A, ...

When he has $N$ of the same system he will then call that version $T$ (TRUE) and the other version $F$ (FALSE).  In the above example, if he has $N$ of version $A$ he will then rewrite the sample path to
T, T, F, T, F, F, T, ...

and otherwise, if he has $N$ of version $B$ he will rewrite the sample path to
F, F, T, F, T, T, F, ...

Hence no matter what, we will have 


*

*$N$ computers with version $T$ and

*$K - N$ computers with version $F$. $\scriptsize (K-N \in [0,N))$


If the first computer was registered as one of the computers with the version that we have $N$ of, i.e. $T$ then the supervisor can announce

We will use {the version on computer 1}.

If the first computer was registered as one of the computers with the version that we have $K-N$ of, i.e. $F$, the employer can tell the supervisor "this is the incorrect version" so the supervisor can announce the other version

We will use {the version different than the one on computer 1}.

By construction the supervisor can always announce the correct version, only checking one computer so $X = 1$, i.e. it is deterministic and not random.
