I'm going to pretty much rephrase my problem in a straightforward terms.
Let's say one has $N$ friends and every one of them has either an odd or even number of cats with probability $0.5 \pm \epsilon$. If $kN$ ($k \in [0;1]$) different friends were randomly chosen and given a one additional cat then how likely is the fact that the probability of having odd\even number of cats is within $[0.5-\delta;0.5+\delta]$ range?
How to evaluate this? I am really confused because this is essentially a probability of some other probability... Originally I wanted to evaluate how distorting can this effect of giving away cats be, but simply calculating the possible probabilities is not enough because e.g. if I had $10$ friends ($5$ of each category) and gave half of them an additional cat then the outcome can be everything from all friends with odd number of cats to all friends with even number of cats.