Can you compute $P(A)$ if you know $P(A|B), P(A|C), P(B)$ and $P(C)$? The agent is described as either loyal or not-loyal. The probability that the agent is loyal is given by ‘p.’ The negotiations can reach two outcomes for the country: favorable and unfavorable. The outcome of the negotiations may give an indication of whether the agent is loyal or not. Suppose that the negotiations have an unfavorable outcome for the country. Suppose that $p($loyal$)=.8$, $p($unfavorable$|$loyal$)=.5$, and $p($unfavorable$|$not loyal$)=.7$. Use Bayes Theorem to calculate the probability that the agent is not loyal. My question is: how can I use Bayes Theorem in this case when I don't know $p($unfavourable$)$ or $p($favourable$)$? Is it even possible to compute $p($not loyal$$unfavourable$)$ from the data given?

The law of total probability says:

$$P(\text{unfavorable})=P(\text{unfavorable}|\text{loyal})P(\text{loyal})+P(\text{unfavorable}|\text{not loyal})P(\text{not loyal})=.5\cdot .8+.7\cdot .2$$

Does this help?