# Bayes theorem for mutations

MEN 2A is a dominant inherited disease caused by a mutation in the RET proto-oncogene. The probability of being sick when you have the mutation of the RET proto-oncogene varies with age and is assumed to be 40% at 40 years of age.

With autosomal dominant inherited diseases, this formula can be used, where p indicated penetration and D is the sick allele:

P(II inherits D given that II is healthy) = $\frac{1-p}{2-p}$

The uncle of the son III1 in the tree has the disease so he assumes it comes from I1 or I2. The father of the son II2 is 40 years old with no symptoms.

What are the odds of the father having the mutation?

What are the odds of the son having the mutation knowing the % in the first question? I just used the formula above as so for the first question: $$\frac{1-0.4}{2-0.4} = 37.5\%$$

I'm having problems with the second though and I assume to use Bayes formula.

I've tried as so:

$$P(\text{son mut given dad mut}) = \frac{P(\text{Dad mut given son mut}) \cdot P(\text{son mut)}}{P(\text{Dad mut})}$$

I know the probability of the dad having the mutation is 37,5% and the probability of the dad having the mutation given that his son has the mutation is 100%. However I can't seem to know what to put in the probability of the son having the mutation? Any help appreciated, thanks! Sorry for a long question.

• We don't know the probability of being sick given that you are homozygous for the mutant allele so we can I guess just consider this to be negligible, so we'll assume that the kid is heterozygote. The allele either come form the dad or the mother with equal probability. the odds are therefore $\frac{0.5}{0.5}=1$. If we were to know that the father (and let's assume the father only) is carrying the mutant allele (let's assume in one copy only), then the probability of the kid to get this allele is 0.5. The odds are $\frac{0.5}{0.5}=1$ again. Did I missunderstand the problem? May 1, 2015 at 18:56

I don't understand your calculations and I don't understand why you're trying to use Bayes formula. I don't know the $\frac{1-p}{2-p}$ formula and I don't understand what it is supposed to calculate. It seems to me that you're overthinking a simple problem.
I neglect the probability that a new mutation occurred and that the son is the first carrier of the mutant in the lineage. I assume that we have a probability of 0 of being sick if we don't carry the disease. Because MEN is a pretty bad disease, the autosomal dominant alelle causing the disease is probably kept at pretty low frequency. From wiki, I read that MEN 2A has a frequency of 1/40000. This means that the probability of being homozygous mutant is $\left(\frac{1}{40000}\right)^2 ≈ 10^{-9}$. I think we can safely assume that the sick kid is heterozygous. Given that the kid is heterozygous, the mutant allele comes either from the mother or from the father with equal probability. So the probability it comes from the father is $0.5$ and the odds are $\frac{p}{1-p}=\frac{0.5}{1-0.5}=\frac{0.5}{0.5}=1$