The question as given reads as follows: "If all of the uncertainties of the premises in the inferences below are equal, how small do these uncertainties have to be to guarantee that the conclusions of these inferences have probability at least .99?"
a) premises: {A,B} Conclusion: A IFF B
b) premises: {A, A implies B} Conclusion: B
Apparently both answers are 0.05(from the back of the book) but I cannot figure out why. Anyone have any idea?
This from the book "probability logic" ernest adams. It is a question in section 3.2.
Maybe there's a typo in your post, but it seems to me that we can do better than $0.05$.
For example, here's (b). We are given that $P(B) \geq 0.99$ and $P(A^c) = P((A \ \text{implies} \ B)^c)$. I am assuming here that the "uncertainty" of $A$ is equal to $P(A^c)$. Also, I will assume that "implies" refers to the material conditional, i.e. $$P(A \ \text{implies} \ B)=P(A^c \cup B),$$ so $$P((A \ \text{implies} \ B)^c) = P(A \cap B^c).$$ Then, by our assumption that the premises have the same uncertainties, it follows that $$P(A^c) = P(A \cap B^c) = P(B^c) - P(B^c \cap A^c) \leq 0.01 \leq 0.05$$ because $P(B) \geq 0.99$ implies $P(B^c) \leq 0.01$ and $P(B^c \cap A^c) \geq 0$.
I leave (a) to you.