Question $2.(20$ Points$)$

Bob would like to classify emails as spam or non$-$spam. He would like to estimate the probability that a new email e containing the keywords $(w_1,w_2,$dots,$w_n)$ is spam by taking all the emails in the training set with those keywords, and then computing the proportion of those emails that are spam. Specifically, he estimates the probability using

$P($ spam $|$ new email $e)=\frac{\text{n}\text{u}\text{m}\text{.}of\text{s}\text{p}\text{a}\text{m}\text{e}\text{m}\text{a}\text{i}\text{l}\text{s}\text{w}\text{i}\text{t}\text{h}\text{k}\text{e}\text{y}\text{w}\text{o}\text{r}\text{d}\text{s}{w}_1,w_2,\text{d}\text{o}\text{t}\text{s}\text{,}{w}_{n}in\text{t}\text{h}\text{e}\text{t}\text{r}\text{a}\text{i}\text{n}\text{i}\text{n}\text{g}\text{s}\text{e}\text{t}}{\text{n}\text{u}\text{m}\text{.}of\text{t}\text{o}\text{t}\text{a}\text{l}\text{e}\text{m}\text{a}\text{i}\text{l}\text{s}\text{w}\text{i}\text{t}\text{h}\text{k}\text{e}\text{y}\text{w}\text{o}\text{r}\text{d}\text{s}{w}_1,w_2,\text{d}\text{o}\text{t}\text{s}\text{,}{w}_{n}in\text{t}\text{h}\text{e}\text{t}\text{r}\text{a}\text{i}\text{n}\text{i}\text{n}\text{g}\text{s}\text{e}\text{t}}.$

Part $($a$)$ Explain why Bob's plan will generally not work. $(5$ Points$)$

Part $($b$)$ Describe the datasets for which Bob's plan might work. Be specific: state which properties are required of the datasets. $(5$ Points$)$