\begin{problem}{An Independence Dad Joke} 
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.

The following three images are possible models involving the genes $G$ and handednesses $H$.

\begin{center}
    \includegraphics[height=0.3\textwidth]{figures/bayesnet1a.png} 
    \hspace{2.5cm}
    \includegraphics[height=0.3\textwidth]{figures/bayesnet1b.png} \\
    \includegraphics[height=0.3\textwidth]{figures/bayesnet1c.png} 
\end{center}


\begin{question}[3]
Which of the three networks above claim that $P(G_{father}, G_{mother}, G_{child}) = P(G_{father})P(G_{mother})P(G_{child})$? Select all that apply.

\qquad\solution{\emptysquare}{\OneAA} (a) \quad 
\qquad\solution{\emptysquare}{\OneAB} (b) \quad 
\qquad\solution{\emptysquare}{\OneAC} (c) \quad 
\end{question}

\begin{question}[3]
Which of the three networks make independence claims that are consistent with the hypothesis about the inheritance of handedness? Select all that apply.

\qquad\solution{\emptysquare}{\OneBA} (a) \quad 
\qquad\solution{\emptysquare}{\OneBB} (b) \quad 
\qquad\solution{\emptysquare}{\OneBC} (c) \quad  
\end{question}

\begin{question}[4]
Which of the three networks is the most direct transcription of the hypothesis?
Choose only one answer.

\qquad\solution{\emptycircle}{\OneCA} (a) \quad 
\qquad\solution{\emptycircle}{\OneCB} (b) \quad 
\qquad\solution{\emptycircle}{\OneCC} (c) \quad 

\end{question}

\end{problem}
