I was wondering how I could calculate the crossover probability of a cascade of $N$ binary symmetric channels. The crossover probability in every channel is $\epsilon$.
I read the result should equal $\epsilon (n)= \frac{1}{2}(1-(1-2\epsilon)^n)$.
Unfortunately, my calculations do not lead that way.
The total number of crossovers follows a Binomial distribution $B(N,\epsilon)$ A global crossover occurs if that number is odd. The probability of that event (well, its complement) is computed here. And it leads to the result you copied.