A simple e-mail spam filter uses Bayes\' theorem to predict whether a given mess

Question

A simple e-mail spam filter uses Bayes' theorem to predict whether a given message is spam or not.

We have a "training" data set of 1,000 e-mails, 500 of which have been identified as spam by a human analyst. The phrase "Instant winner" appears in 460 of the spam messages, and 15 of the non-spam messages.

For the internet as a whole, 90% of all e-mails are estimated to be spam messages. If a new e-mail arrives containing the phrase "Instant winner", what is the probability that it is a spam message?

(Hint: use Bayes' theorem, and be careful when you calculate P("instant winner")! Our sample of 1,000 e-mails isn't representative of the fact that 90% of all e-mails are spam...)

Explanation / Answer

Ans:

P(spam)=500/1000=0.5

P(non spam)=1-0.5=0.5

P(instant inner/spam)=460/500=0.92

P(instant winner/non spam)=15/500=0.03

P(spam/instant winner)=P(instant winner/spam)*P(spam)/[P(instant winner/spam)*P(spam)+P(instant winner/non spam)*P(non spam)]

=0.92*0.5/[0.92*0.5+0.03*0.5]

=0.92/0.95

=0.9684

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