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Bayesian Statistics: Find the posterior odds

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The human X chromosome carries gene that is essential for normal clotting of the blood. The defect in this gene that is responsible for the blood disease hemophilia is recessive: no disease develops in women who have at least one X chromosome with a normal gene. However, a man whose X chromosome has the defect gene develops the disease. As a result, hemophilia occurs almost exclusively in males who inherit the gene from non-hemophiliac mothers.

Sally has a brother with hemophilia, but neither she, her parents, nor her two sons (ages 5 and 8) have the disease. Let Ho be the null hypothesis that Sally does not carry the hemophilia gene, Ha be the alternative hypothesis that Sally does carry the hemophilia gene. Assign a prior distribution P(Ho)=P(Ha)=1/2. Let Y denote the event that Sally's two sons are healthy.

Find the posterior odds
P(Ho|Y)/P(Ha|Y). Does this provide evidence for Ho or Ha?

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The expert examines Bayesian statistics for posterior odds.

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Answer: Given

Now

(Note that given both ...

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