I am stuck at the following exercise:
Let $\mathcal{C} = (\mathcal{X} , P, \mathcal{Y})$ be a channel with $\mathcal{X} = \mathcal{Y} = \{0,1\}$ and transition matrix
$$ P = \begin{bmatrix} 7/8 &1/8 \\ 1/8 &7/8 \end{bmatrix}.$$
The input probabilities are $P[X = 0] = 0.3, P[X = 1] = 0.7$. Each input bit is encoded by transmitting it through the channel $2n + 1$ times, and the decoding function is the majority bit (the bit $0$ or $1$ that appears more times in the sequence of $2n + 1$ output bits). Let $f(n)$ be the expected probability of error, that is, the probability that the decoded bit is different from the encoded bit.
Express $f(n)$ as a sum of the form $f(n) = \sum_{k=\ldots}^\ldots (\ldots)$.
I do not know what is meant with the "expected probability of error", we did not define this in the lecture. Could you explain this to me?