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
0.10⨉ 0.95
.10⨉0.95 + 0.90⨉0.11
=0.4896
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