Covid swab, false positives and Bayes

Problem

A swab test is tried on $333$ healthy people, giving $85$ positives, and on $490$ sick people, giving $29$ negatives. The disease incidence is $P(A)=0{,}01$. a) Find $P(n\mid A)$, $P(p\mid S)$ and from them $P(p\mid A)$, $P(n\mid S)$. b) Find the probabilities of the elementary cases $Ap,An,Sp,Sn$ and the total probabilities $P(p)$ and $P(n)$. c) Carlo is positive: what is his probability of being healthy? Giorgia is negative: what is her probability of being sick?

Solution

a) $P(p\mid S)=\dfrac{85}{333}\approx0{,}2553$, $P(n\mid A)=\dfrac{29}{490}\approx0{,}0592$, hence $P(p\mid A)=0{,}9408$, $P(n\mid S)=0{,}7447$. b) With $P(A)=0{,}01$, $P(S)=0{,}99$: $P(Ap)=0{,}009408,\ P(An)=0{,}000592,\ P(Sp)=0{,}25275,\ P(Sn)=0{,}73725;$ $P(p)=P(Ap)+P(Sp)\approx0{,}2622,\qquad P(n)=P(An)+P(Sn)\approx0{,}7378.$ c) $P(S\mid p)=\dfrac{P(Sp)}{P(p)}=\dfrac{0{,}25275}{0{,}2622}\approx0{,}9641$; $\;P(A\mid n)=\dfrac{P(An)}{P(n)}=\dfrac{0{,}000592}{0{,}7378}\approx0{,}00080$. $\boxed{\,P(S\mid p)\approx0{,}964,\qquad P(A\mid n)\approx0{,}0008\,}$