Bayes Rule Disease diagnosis

Question

I'm having trouble understanding the answer to the following Bayes rule question

I have a few questions about this:

1. I know that the the prompt provides comparisons between H1N1 and fever, flue and fever, and neither and fever. But why does it make sense to compare Pr(H1N1|Fever), Pr(Flue|Fever), Pr(Neither|Fever) to answer the question, "Is it more likely that you have H1N1, the flue, or neither ?" Fever could have been replaced with some other common condition right, like Pr(H1N1|wear blue socks), as long as that condition was given for everyone?

2. Using Bayes rule and the total law of probability I set up the problem like this:

$Pr(A|D) = \frac{Pr(A \cap D)}{Pr(D)}$

I understand that we can replace $Pr(A \cap D)$ with $Pr(D|A)Pr(A)$ using Bayes rule.

$Pr(D) = Pr(A \cap D) + Pr(\overline{A} \cap D)$ according to the law of total probability.

So why is it that I can't do...

$Pr(D) = Pr(D|A)Pr(A) + Pr(D|A)(1 - Pr(A))$

but instead have to sum up the probabilities of all the other scenarios?

Answer 1

1. Yes, it's possible to consider something else, like wearing blue socks. However, in the problem you have a high fever. In other words, having a fever is given.

2. Sure you can do $P(D) = P(D|A)P(A) +P(D|\bar A)P(\bar A)$, but the problem is that $P(D|\bar A)$ is not readily available. It wasn't given in the problem. If you try working it out, you end up partitioning over the scenarios anyway.