Let us suppose we are transmitting $n_1$ zeros, each encoded with a $[n_2,1,n_2]$ repetition code, with odd $n_2$, so obtaining a codeword of length $n=n_1n_2$. What is the probability that, after decoding, there are $w$ erred bits, if the channel introduces EXACTLY $0\leq t\leq n$ errors? I assume that a bit is evaluated incorrectly if the corresponding block of $n_2$ symbols contains less $0$s than $1$s.
If the channel was a BSC, I could associate a probability of error to each bit and treat each bit independently from the others. In this case I cannot assume that any bit is erred with probability $\frac{t}{n}$.
What I tried
The only way I see to calculate this probability is $$ Pr\{w \hspace{2mm} \mathrm{errors} \} = \frac{\sum_{i_0=\lceil\frac{n_2}{2}\rceil}^{n_2}\ldots\sum_{i_w=\lceil\frac{n_2}{2}\rceil}^{n_2} \sum_{j_0=0}^{\lceil\frac{n_2}{2}\rceil-1}\ldots \sum_{j_{n_1-w}=0}^{\lceil\frac{n_2}{2}\rceil -1} \binom{n_2}{i_0} \ldots \binom{n_2}{i_w} \binom{n_2}{j_0} \ldots \binom{n_2}{j_{n_1-w}} } {\binom{n}{t}} $$
with $i_0+\ldots+i_w+j_0+\ldots+j_{n_1-w}=t$.
This requires listing of all the possible partitions and seems overcomplicated.
Any better ideas?