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Estimates of the prevalence of Alzheimer's disease have recently been provided by Pfeffer et al. [8]. The estimates are given in Table \(3.5 .\) Suppose an unrelated 77 -year-old man, 76 -year-old woman, and 82-year-old woman are selected from a community.Suppose we know two of the three people have Alzheimer's disease. What is the conditional probability that they are both younger than 80 years of age?$$\begin{array}{lcc}\hline \text { Age group } & \text { Males } & \text { Females } \\\\\hline 65-69 & 1.6 & 0.0 \\\70-74 & 0.0 & 2.2 \\\75-79 & 4.9 & 2.3 \\\80-84 & 8.6 & 7.8 \\\85+ & 35.0 & 27.9 \\\\\hline\end{array}$$

Suppose that a disease is inherited via an autosomal recessive mode of inheritance. The implications of this mode of inheritance are that the children in a family each have a probability of 1 in 4 of inheriting the disease.What is the probability that exactly one sibling is affected?

The familial aggregation of respiratory disease is a wellestablished clinical phenomenon. However, whether this aggregation is due to genetic or environmental factors or both is somewhat controversial. An investigator wishes to study a particular environmental factor, namely the relationship of cigarette-smoking habits in the parents to the presence or absence of asthma in their oldest child age 5 to 9 years living in the household (referred to below as their offspring). Suppose the investigator finds that (1) if both the mother and father are current smokers, then the probability of their offspring having asthma is \(.15 ;(2)\) if the mother is a current smoker and the father is not, then the probability of their offspring having asthma is \(.13 ;(3)\) if the father is a current smoker and the mother is not, then the probability of their offspring having asthma is \(.05 ;\) and (4) if neither parent is a current smoker, then the probability of their offspring having asthma is .04.Suppose the smoking habits of the parents are independent and the probability that the mother is a current smoker is \(.4,\) whereas the probability that the father is a current smoker is .5. What is the probability that both the father and mother are current smokers?

A dominantly inherited genetic disease is identified over several generations of a large family. However, about half the families have dominant disease with complete penetrance, whereby if a parent is affected there is a \(50 \%\) probability that any one offspring will be affected. Similarly, about half the families have dominant disease with reduced penetrance, whereby if a parent is affected there is a \(25 \%\) probability that any one offspring will be affected. Suppose in a particular family one parent and two of the two offspring are affected.What is the probability that the mode of transmission for this particular family is dominant with complete penetrance? Is this a prior probability or a posterior probability?