Conditional probability with a set of complementary events Let $\mathbb{P}(b\mid a)=1$, from this it follows that $\mathbb{P}(a\mid\bar{b})=0$.
If $\mathbb{P}(b\mid a)=0.9$, what can be said about $\mathbb{P}(a\mid\bar{b})$?
Generalizing the problem, consider $\mathbb{P}(b_i\mid a)$ is known for some set of events $\{b_i\}$, what type of bounds can be calculated for $\mathbb{P}(a\mid\bar{b}_1\cap\bar{b}_2\cap\cdots\cap\bar{b}_k)\vphantom{\dfrac11}$?
 A: $$\displaystyle \mathbb{P}(b|a)=\frac{\mathbb{P}(b\cap a)}{\mathbb{P}(a)}.$$
Assuming $\mathbb{P}(a)\not=0$, $\mathbb{P}(b|a)=1$ if and only if $\mathbb{P}(b \cap\ a)=\mathbb{P}(a)$.
$$\displaystyle \mathbb{P}(a|\bar{b})=\frac{\mathbb{P}(a\cap \bar{b})}{\mathbb{P}(\bar{b})}.$$
Assuming $\mathbb{P}(\bar{b})\not=0$, then $\mathbb{P}(a |\bar{b})=0$ if and only if $\mathbb{P}(\bar{b} \cap a)=0$.
Assuming again that $\mathbb{P}(a)\not=0, \mathbb{P}(\bar{b})\not=0$, we can make the following conclusions:

$$\mathbb{P}(b\cap a)= \mathbb{P}(a) \iff \mathbb{P}(a)-\mathbb{P}(b \cap a)=0 \iff \mathbb{P}(a)(1 - \mathbb{P}(b|a))=0 \iff \\
 \mathbb{P}(a)(\mathbb{P}(\bar{b}|a))=0 \iff \mathbb{P}(\bar{b}|a)=0 \iff \frac{\mathbb{P}(\bar{b}\cap a)}{\mathbb{P}(a)}=0
 \iff \mathbb{P}(\bar{b}\cap a)=0 \iff \\ \frac{\mathbb{P}(\bar{b}\cap a)}{\mathbb{P}(\bar{b})}=0 \iff  \ \mathbb{P}(a|
 \bar{b})=0.$$

Hence, none of the statements you mention generalize; what appear to be results regarding $\mathbb{P}(b|a)$ and $\mathbb{P}(a|\bar{b})$ are actually fundamentally only statements regarding:
$$\mathbb{P}(a\cap b)\ \text{and}\ \mathbb{P}(a\cap \bar{b})$$
using the Law of Total probability.
For the statements you have, we need in general to know the values of $\mathbb{P}(a)$ and $\mathbb{P}(\bar{b})$. Since that information is not contained in the statements you have given, we cannot say anything more. 
Thus, even though we can generalize the application of the Law of Total Probability for analysis of the probabilities $\mathbb{P}(a \cap b)$ and $\mathbb{P}(a \cap \bar{b})$, we cannot do so for $\mathbb{P}(b|a)$ and $\mathbb{P}(a|\bar{b})$ without additional information.
Regarding the questions of bounds, whatever inferences we can make would have to account for the uncertainty of $ 0 \le \mathbb{P}(a) \le 1$ and $0 \le \mathbb{P}(\bar{b})\le 1$. While these would lead to bounds on the possible results of the values, they would be very imprecise and hence not very useful.
