Let x $=($x$1,$ x$2,23)$ describe a patient's medical record, where each x in $\{0,1\}.($For example, each x could record the binary outcome of a certain lab test$).$ The label y indicates the presence or absence of a certain disease, with y in $\{0,1\}.$ Suppose the prior distribution of Y is P$($Y $=0)=0\mathrm{.}8,$ and P$($Y $=1)=0\mathrm{.}2,$ and the likelihood is P$($X$=($x$1,$ x$2,$ x$3)|$Y $=$ y$)=3-1(0\mathrm{.}5+0\mathrm{.}25)(0\mathrm{.}75-0\mathrm{.}5)1-2.$ If h is the predicted outcome while the truth is y$,$ the loss l$($h$,$ y$)$ incurred is given by l$(0,0)=0,$ l$(0,1)=+1000,$ l$(1,0)=+100,$ and $(1,1)=-500.($i$)$ Determine the Bayes Optimal hypothesis h Bayes$-$Optimal.

$($ii$)$ What is the Bayes Optimal Risk?