Given the number of individuals who have a recessive disorder, how can one calculate the probability that the disorder was inherited?
Assuming...


The parents are the real biological parents (no cheating allowed!)
There is a purely dominance/recessivity relationship between the alleles and no individual can express the disorder without being homozygous for the disease causes allele
De novo mutations for the disorder are rare enough to be ignored in the calculation of this probability
Trisomy and monosomy is not involved


, then the two parents must necessarily be carrier (both of them) as long as at least one of their children express the disease.

Note, by the way that some of the above assumptions are too strict and could be partially released.



This question does not require any complicated bayesian statistics (or any statistics at all) or any use of Hardy-Weinberg rule. If a children expresses a purely recessive disease, then (s)he is necessarily aa, where a is the recessive allele and A is the dominant allele(s). It means that both parents must have at least one a. Because, no parents express the disease, both parents are necessarily Aa (or aA)