Recall the probabilistic model that we described in class for the binary random variables $\{$E $=$ Earthquake,

B $=$ Burglary, A $=$ Alarm, J $=$ JohnCalls, M $=$ MaryCalls$\}.$ We also expressed this model as a belief

network, with the directed acyclic graph $($DAG$)$ and conditional probability tables $($CPTs$)$ shown below:

Alarm

Earthquake Burglar

John Calls Mary Calls

P$($E$=1)=0\mathrm{.}002$ P$($B$=1)=0\mathrm{.}001$

P$($A$=1|$E$=0,$B$=0)=0\mathrm{.}001$

P$($A$=1|$E$=0,$B$=1)=0\mathrm{.}94$

P$($A$=1|$E$=1,$B$=0)=0\mathrm{.}29$

P$($A$=1|$E$=1,$B$=1)=0\mathrm{.}95$

P$($J$=1|$A$=0)=0\mathrm{.}05$

P$($J$=1|$A$=1)=0\mathrm{.}90$

P$($M$=1|$A$=0)=0\mathrm{.}01$

P$($M$=1|$A$=1)=0\mathrm{.}70$

Compute numeric values for the following probabilities, exploiting relations of conditional independence as

much as possible to simplify your calculations. You may re$-$use numerical results from lecture, but otherwise

show your work. Be careful not to drop significant digits in your answer.

$($a$)$ P$($E $=1|$A$=1)($c$)$ P$($A$=1|$M $=1)($e$)$ P$($A$=1|$M $=0)$

$($b$)$ P$($E $=1|$A$=1,$ B $=0)($d$)$ P$($A$=1|$M $=1,$ J $=0)($f$)$ P$($A$=1|$M $=0,$ B $=1)$

Consider your results in $($b$)$ versus $($a$),($d$)$ versus $($c$),$ and $($f$)$ versus $($e$).$ Do they seem consistent with

commonsense patterns of reasoning?