As communication security becomes more and more a problem, when a message is received, it must authenticated, by using a secret enciphering key. Sometimes, though, it can fall into wrong hands, thus allowing an unauthentic message to appear to be authentic. Assume that 80% of all messages received are authentic. Furthermore, assume that only 5% of all unauthentic messages are sent using the correct key and that all authentic messages are sent using the correct key. Find the probability that a message is authentic given that the correct key is used.
How do I compute this event, I've been scratching my head for the last half an hour and tried to look at the problem from different angles, I tried thinking of it as $ P(A/K) $ $A$ - message is authentic and $K$ - key is correct and then tried to compute with Bayes but it makes no sense since $P(K)$ is always going to be 1 since it is specified that the identified key is authentic and $P(K/A)$ would also result in 1, since the message is authentic the key will always be correct. Am I even making sense? I feel like I'm missing something right under my nose because at first glance it doesn't seem to be such a difficult problem, so I would appreciate some help...
Let $A \equiv \text{{Message is authentic}}, K \equiv \text{{Message sent with correct key}}$.
You want $$P(A|K) = \frac{P(K|A)P(A)}{P(K)}$$
You are correct that $P(K|A) = 1$ because $A \subseteq K$. However $P(K) \neq 1$. To find $P(K)$ we can use the law of total probability.
Note that $A$ and $\overline{A}$ partition the sample space. Then we have that
\begin{align} P(K) &= P(K|A) P(A) + P(K|\overline{A}) P(\overline{A})\\ &= 1\cdot 0.80 + 0.5\cdot0.20\\ &= 0.90 \end{align}
Then finally we have
$$P(K) = 8/9$$