I have the following problem. I have a vector of size $N$ in $\mathbb{F}_2$ containing exactly $m$ zeros and $n$ ones with $m>n$. Then, a random noise is applied on each bit independently such that with probability $p$ the value of a bit is changed.
I'm trying to bound the probability that the majority value changes, i.e. that after applying the noise we get more ones than zeros.
What I did is the following: I created a random variable $X$ denoting the number of zeros that are modified by the noise and $Y$ the number of ones that are modified. I was then able to bound the probability by summing over all $\Pr[X>k \cap Y < N/2-m+k]$ such that $n+X-Y > N/2$.
However, this bound is rather complicated and I'm looking for simpler one. Any idea?