Question 1061858
I suspect that the answer may end up being almost the same, but
why do you start assuming that you have a population where
180 people are infected and 99800 are uninfected?
That would be a total of {{{180+99800=99980}}} ,
and the ratio of infected to total would be
{{{180/99980=18/9998=approximately}}}{{{18/9999=2/1111=1/555.5}}} .
That means 1 person out of 555.5 people would be infected,
and that counts as a mistake.
 
You could have started with 100,000 people.
Of those, {{{100000/500=200}}} are infected,
and the remaining {{{100000-200=99800}}} are uninfected.
Of the {{{200}}} infected people,
{{{0.1(200)=20}}} will test negative.
Of the {{{99800}}} uninfected,
{{{0.9(99800)=89820}}} will test negative, as you said.
The total number of people testing negative is
{{{20+89820=89840}}} .
Comparing to that number, the number of uninfected people is
{{{89820/89840=0.999777}}} (rounded), which could be stated as {{{highlight("99.98%")}}} .
 
NOTES:
With your calculation you should get
{{{89820/89838=0.999800}}} (rounded), which could be stated as 99.98%.
You mistakenly wrote your final calculation as
{{{89820/89832=0.999866}}} (rounded), which could be stated as 99.99%.
Neither of those numbers can be rounded to 99.9%;
they would round to 100.0% if we only give them one decimal digit.
 
Note that before the test,
a person knew that the probability of being uninfected was
{{{499/500=0.998000}}} or 99.80%.
If this virus is life-threatening (or life-altering),
people may want to know more.
If they test positive, maybe there is something they can do about it.
If they test negative,
going from 99.80% to 99.98% sure that they are safe may be cause for celebration.