$(8$ points$)$ Bayes Classifier

An e$-$commerce company has launched an online marketing

campaign

using banner advertisements placed on various third$-$party web

sites to

drive visitors to the company's web site. The web site has

received $50,000$

hits since the campaign was launched, out of which $10,000$ of

them were

the result of click$-$throughs of the banner advertisement.

Furthermore,

$20\%$ of the visitors directed to the Web site via the banner

advertisement

bought an item at the web site $($compared to only $6\%$ of the

visitors

arriving at the web site without clicking on a banner

advertisement who

ended up buying an item$).$ Based on this information:

Calculate the probability that a visitor will buy an item at

the

website $($regardless of how the visitor arrives at the website$).$

Determine whether the advertisement campaign is

successful. To

do this, you need to check the following: among the visitors who

made a purchase at the web site, are they more likely to arrive at

the web site via the banner advertisement or arrive without

clicking

on the banner? Show your calculations clearly.