Probability and genetics - Bayes' theorem Britney can be homozygous $HH$ or heterozygous $Hh$ with equal probability.
Hemophilia is a mostly inherited genetic disorder. A test to detect a dominant allele $h$, responsible for the disorder, is carried out.
The test has $85\%$ reliability in heterozygous women (with $Hh$ genotype), that is, it successfully detects the presence of the allele $h$ in $85\%$ of the cases, while in homozygous women (with $HH$ genotype) it fails to detect $h$ in $1\%$ of the cases.
We want to calculate the following probabilities:
$P (\text{Britney}\,Hh | \text{test was positive})$
and
$P(\text{Britney}\,HH | \text{test was negative})$
I am not sure for the correct interpretation of the question, as I had to translate some terms I am not familiar with.
With the little knowledge I have on statistics, I will make an attempt:

Prior probability Britney is homozygous or heterozygous $P(ΗΗ)= P(Hh) = 0.5$

$$P(E|Hh)= \text{Probability of a Positive Test Result given Britney is Heterozygous} = 0.85\\
\text{So, we have}\\
P(E|HH)= \text{Probability of a Positive Test Result given Britney is Homozygous} = 0.15$$
We want $$P(HH|E) = \text{Probability of Britney being Heterozygous given the test yields a Positive Result}$$
We also want $$P(Hh|E^c) = \text{Probability of Britney being Homozygous given the test yields a Negative Result}$$
So for a)
$$P(HH|E) = {P(E|HH) P(HH) \over P(E)} = {P(E|HH) P(HH) \over P(E|HH)P(HH) + P(E|{Hh}) P({Hh})}$$
and similarly for the second. Are these correct?
EDIT: Can you tell me if this is correct?
"$P(E|HH)= \text{Probability of a Positive Test Result given Britney is Homozygous} = 0.15$"
or is it
"$P(E|HH)= \text{Probability of a Negative Test Result given Britney is Heterozygous} = 0.15$"?
 A: As I understand from your setting, you could approach the problem as follows:
Suppose you let $T$ be the event "the test is positive" (so the test detects the allele $h$), $HH$ the event "homozygous genotype", and $Hh$ the event "heterozygous genotype". We wish to compute the probabilities $Pr(Hh|T)$ and $Pr(HH|T^c)$.
We know the following:

*

*$Pr(HH)=Pr(Hh)=1/2$, so the probability of having either genome is the same.

*$Pr(T|Hh)=0.85$ from which you can deduce that $Pr(T^c|Hh)=0.15$.

*$Pr(T^c|HH)=0.01$ from which you can deduce that $Pr(T|HH)=0.99$.

From there, using Bayes' theorem, the law of total probability and remembering that the probability of having either genome is equal, we can directly compute
\begin{align}
Pr(Hh|T)=\frac{Pr(T|Hh)Pr(Hh)}{Pr(T)}
&=\frac{Pr(T|Hh)Pr(Hh)}{Pr(T|Hh)Pr(Hh)+Pr(T|HH)Pr(HH)}\\
&=\frac{0.85\cdot 1/2}{(0.85+0.99)/2}\approx 46.2\%,
\end{align} and
\begin{align}
Pr(HH|T^c)=\frac{Pr(T^c|HH)Pr(HH)}{Pr(T^c)}&=\frac{Pr(T^c|HH)Pr(HH)}{Pr(T^c|HH)Pr(HH)+Pr(T^c|Hh)Pr(Hh)}\\
&=\frac{0.01\cdot 1/2}{(0.01+0.15)/2}=6.25\%.
\end{align}
Edit: I've confused the original statement and so had wrong calculations. The procedure however stays the same.
A: 
By the law of total probability, $$P(Positive)=P(Positive|HH)P(HH)+P(Positive|Hh)P(Hh)=0.5\cdot0.99+0.5\cdot0.85=0.5(0.99+0.85)=0.5\cdot1.84=0.92$$ and so P(Negative)=0.08 (can also be calculated from the law of total probability).
(a)By Bayes Theorem, $$P(Hh|Positive)=\frac{P(Hh\cap Positive)}{P(Positive)}$$
$$=\frac{0.85\cdot 0.5}{0.92}=\frac{85}{184}≈0.4619=46.19\% $$
 Similarly, (b) By Bayes Theorem, $$P(HH|Negative)=\frac{P(HH\cap Negative)}{P(Negative)}$$
$$=\frac{0.01\cdot 0.5}{0.08}=\frac{1}{16}=0.0625=6.25\%$$
