$0\mathrm{.}4\%$ find the joint probability distribution$$

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P$($Cavity V Toothache$)=$ P$($Cavity OR Toothache$)=$

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Hint: Remember P$($A V B$)=$ P$($A$)+$ P$($B$)-$ P$($A $$ B$)$ in case if A and B are not independent. If A and B are completely independent P$($A $$ B$)=0$

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Answer the following questions $($up to $3$ decimal points$)$:$$

$(1\%)-$ What is the probability of having Covid and getting a positive PCT test?$$

P$(+$c $,+$p$)=$

Hint: Please review variable elimination slides as there is a variable here that needs to be eliminated $($i$.$e$.,$ summed out$)$ to calculate the above probability$$

$(1\%)-$ What is the probability of not having Covid and getting a positive PCR test?$$

P$(-$c $,+$p$)=$

Same hint as above applies$$

$(1\%)-$ Conditional probability review $$ What is the probability of having Covid given that the PCR test has been positive$$

P$(+$c $|+$p$)=$

Hint: Conditional probability slides and formula review$$

$(1\%)-$ Bayes theorem review: What is the probability of having Covid given that the Antigen test has been positive?$$

P$(+$c $|+$a$)=$

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$[1\%]$ Consider the following Bayes Net structure$$

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List at least $4$ conditional independences from the above structure $($e$.$g$.,$ A

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$(0\mathrm{.}4\%)-$ L $$ B $-$ True or False$$

$(0\mathrm{.}4\%)-$ L $$ T $|$ E $-$ True or False$$

$(0\mathrm{.}4\%)-$ L $$ K $|$ E $-$ True or False$$

$(0\mathrm{.}4\%)-$ T $$ K $|\{$E$,$ X$\}-$ True or False$$

$(0\mathrm{.}4\%)-$ B $$ M $|$ K $-$ True or False