In information theory, how do I calculate the probability of an erroneous transmission? Let's take for instance a binary symmetric channel with an error probability $ 1-d=0.25 $ and send codewords of length 6 coded in a Hamming code able to correct up to 1 error.
We assume independence of bit errors. This is a somewhat dubious assumption, since errors often occur in bursts.
The probability of erroneous interpretation (or inability to decode) of a codeword of length $6$ is the probability that $2$ or more bits are incorrectly transmitted. The probability that $0$ bits are wrong is $(0.75)^6$. The probability that exactly $1$ bit is wrong is $6(0.25)(0.75)^5$. Add these two numbers, subtract the result from $1$.