Let $H_x$ be a random variable denoting the handedness of an individual $x$, with possible values $l$ or $r$. A common hypothesis is that left- or right-handedness is inherited by a simple mechanism; that is, perhaps there is a gene $G_x$, also with values $l$ or $r$, and perhaps actual handedness turns out mostly the same (with some probability $s$) as the gene an individual possesses. Furthermore, perhaps the gene itself is equally likely to be inherited from either of an individual’s parents, with a small nonzero probability $m$ of a random mutation flipping the handedness.
1. Which of the three networks in Figure handedness-figure claim that $ {\textbf{P}}(G_{father},G_{mother},G_{child}) = {\textbf{P}}(G_{father}){\textbf{P}}(G_{mother}){\textbf{P}}(G_{child})$?
2. Which of the three networks make independence claims that are consistent with the hypothesis about the inheritance of handedness?
3. Which of the three networks is the best description of the hypothesis?
4. Write down the CPT for the $G_{child}$ node in network (a), in terms of $s$ and $m$.
5. Suppose that $P(G_{father}l)=P(G_{mother}l)=q$. In network (a), derive an expression for $P(G_{child}l)$ in terms of $m$ and $q$ only, by conditioning on its parent nodes.
6. Under conditions of genetic equilibrium, we expect the distribution of genes to be the same across generations. Use this to calculate the value of $q$, and, given what you know about handedness in humans, explain why the hypothesis described at the beginning of this question must be wrong.