In AIMA 2nd edition, there is this exercise
13.8 After your yearly checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease and that the test is 99% accurate (i.e., the probability of testing positive when you do have the disease is 0.99, as is the probability of testing negative when you don't have the disease). The good news is that this is a rare disease, striking only 1 in 10,000 people of your age. Why is it good news that the disease is rare? What are the chances that you actually have the disease?
I solved it no problem, but the problem solution manual did this
We got the same result at the end, but p(test|~disease) is not the compliment of p(test|disease), how this unfolded the same result?
Edit 1:
Why p(test|~disease) = 0.01 ? I know that p(test|disease)=0.99 but those two probabilities do not complement each other, yet p(test|~disease) was replaced by 0.01 without any explanation?
Let ${}^c$ denote the complement. Recall that \begin{align*} \mathbb{P}(A) &= \mathbb{P}(A \cap B) + \mathbb{P}(A \cap B^c) \\ &= \mathbb{P}(A \, | \, B)\mathbb{P}(B) + \mathbb{P}(A \, | \, B^c)\mathbb{P}(B^c) \end{align*} for any set $B$ with $\mathbb{P}(B) \in (0,1)$.