If you dont like Bayes theorem then use this
HIV+ 10% of population
then HIV- is 90% of population
the kit correctly identifies HIV+ in 95% of time
the kit incorrectly identifies HIV- in (100-89)%= 11% of time
So, applying Baye's theorem
.10⨉0.95 + 0.90⨉0.11
For creating automatas you can also ...