Does shift cipher with non-uniform plain text probability have perfect secrecy?
For any ciphertext $c \in C$ and message $p \in P$ we have
$$P(c|p) = \frac{P(p\cap c)}{P(p)} = \frac{P(p\cap k)}{P(p)},$$
where $k = c-p$. This is because if the plaintext is $p$, the ciphertext is $c$ if and only if the key is $k$. Now, if the key is chosen independently of the plaintext we have $P(p\cap k) = P(p)\cdot P(k)$, and so finally $P(c|p) = P(k) = 1/8$ if the key is chosen uniformly.