IV$.$ Evaluation $(10$ pts$)$

Suppose that you are working for a phone company that wants to predict churn. Churn is when customers cancel their subscriptions. Let "actual" be a binary random variable meaning that the customer actually churns, and let "flag" be a binary random variable meaning that a classifier predicts that the customer will churn. Note that actual $=1,$ the churn base rate, is the probability that a customer would actually churn. This can be represented by the fraction of customers with "actual" value of $1.$ Similarly, $P(flag=1)$ is the fraction of customers that are predicted to churn. We know that the churn base rate is $5\%.$ We define:

$a=P($ actual $=1|$ flag $=1)$

$b=P($ actual $=0|$ flag $=0)$

You have hired Dr$.$ Lin as a data mining consultant. He claims that he can train a classifier that will achieve $a=0\mathrm{.}8$ and $b=0\mathrm{.}9.$

$($a$)$ Drawn a confusion matrix. Show how to define actual $=1$ and flag $=1$ in terms of the entries in the confusion matrix $($TP$,$ TN$,$ FP$,$ FN$).$

$($b$)$ Show how to define a and b in terms of the entries in the confusion matrix $($TP$,$ TN$,$ FP$,$ FN$).$

$($c$)$ Prove Dr$.$ Lin wrong, because no such classifier exists.