Negating conditional probability I'm refreshing on bayes theorem and conditional probability and I ran across these practice problems. I was trucking along until problem 9, which states:
HIV The New York State Health Department reports a 10% rate of the HIV virus
for the “at-risk” population. Under certain conditions, a preliminary screening test
for the HIV virus is correct 95% of the time. (Subjects are not told that they are
HIV infected until additional tests verify the results.) If someone is randomly
selected from the at-risk population, what is the probability that they have the
HIV virus if it is known that they have tested positive in the initial screening?

Seems pretty straight forward. Let P(H) denote the probability of having HIV.
Then,
$P(H) = .1$
$P(~H) = .9$
$P(+ | H) = .95$
and we need to find
$P(H | +) = \frac{P(H)P(+ | H)}{[(P(H)P(+ | H)] + [P(~H)P(+ | ~H)]}$
After running the numbers, they somehow conclude that
$P(+ | ~H) = 1 - P(+ | H) = .05$
I figured this out by working backwards from their answer.
The problem is I have no idea how they reached this conclusion. Doing what they did above they should be including both $P(- | H)$ and $P(+ | ~H)$ together since these are the cases that are the opposite of $P(+ | H)$.
The reason I'm confused is this doesn't seem like the proper way to refer to the probability that they tested positive given that they don't have HIV, strictly because what I suppose they did above also includes the case where they tested negative even though they have HIV.
Can someone walk me through their logic here?
 A: Let $H$ denote the event "has HIV". Let $+$ denote the event "tests positive in initial screening".

HIV The New York State Health Department reports a 10% rate of the HIV virus
for the “at-risk” population.

$P(H)=0.1$

Under certain conditions, a preliminary screening test
for the HIV virus is correct 95% of the time.

This statement implies both that $P(+|H)=0.95$ and $P(-|\overline{H})=0.95$; if only one of them were 0.95, they'd have to talk about either sensitivity or specificity, rather than accuracy (the intended meaning of 'accuracy' should probably have been made explicit, but as a plain-English phrase, for it to be accurate it must have some level of performance under both $H$ and $\overline{H}$) -- the implication is then that a single number for accuracy would imply it holds for both; while related to the technical meaning in binary classification, I don't think they intended exactly that, but rather the more general sense of 'measuring what it's supposed to measure').
In short, the question could be less ambiguous than it is.
Edit: Note Alexis points out in comments that some definitions of test accuracy don't require them be the same; in that case you'd presumably be working with a specific definition of accuracy and would know exactly how that combines the two.
A: So I think the question is actually quite poorly worded. First, when they say that the preliminary screening test is correct 95% of the time, I believe that it correctly should be interpreted as the following: P(+|H) + P(- | ~H) = .95. Basically, it's correct for true positives and true negatives 95% of the time. 
In the set up of this problem however, the author wasn't too clear and confused what "correct diagnoses" actually means. You picked up on that during your opposite of P(+|H) :-).
A: Okay, so first step is to label what we know
1. 

$P($${infected}$) $= 0.10$ 
$P($$\overline{infected}$) $= 0.90$ 

$P($${positive}$) $= 0.95$ 
$P($$\overline{positive}$) $= 0.05$


2.


Once we have what we know, most people will think to create a table similar to this:





Positive
Negative (Not Positive)
Totals




Infected
9,500
500
10,000


Not Infected
85,500
4,500
90,000



95,000
5,000
100,000





However, this is WRONG !!!


3.


Your table needs to have the structure of true-positive, false-positive, true-negative, and false-negative.


A true positive is an outcome where the model correctly predicts the positive class.

Similarly, a true negative is an outcome where the model correctly predicts the negative class.
A false positive is an outcome where the model incorrectly predicts the positive class.

And a false negative is an outcome where the model incorrectly predicts the negative class.


The table looks such as :




4.


Your table should look like this:





Positive
Negative (Not Positive)
Totals




Infected
9,500
500
10,000


Not Infected
4,500
85,500
90,000



14,000
86,000
100,000






5.


If someone is randomly selected from the
at-risk population, what is the probability that they have the HIV virus if it is known that they have tested positive in the initial screening?


= 9500/14,000

= 0.6785714286

= 0.679
