Question

When e-mails are sent a spam filter typically determines if the e-mail contains words on a list that are designated as spam. In this case the e-mail does not go into our inbox. Suppose that the probability that an e-mail contains a word on the spam list is 3/8. Further, suppose that the probability of an e-mail being spam is 5/7. If we receive an e-mail in our inbox, what is the probability that it is not spam

Answer 1

Let I and S denote the events that the email is received in inbox and that the email is spam respectively

Probability that the email is received in inbox

P(I) = 1 - P(email contains a word on the spam list) = 1 - (3/8)

= 5/8

P(S) = 5/7

Probability that an email is received in inbox and that email is spam,

P(I and S) = P(S) - P(email is designated as spam)

= 5/7 - 3/8 = 19/56

P(I | S) = P(I and S)/P(S) = 19/40

Now, Probability that the email we receive in our inbox is not spam,

P(S' | I) = 1 - P(S | I) = 1 - P(I | S) * P(S)/P(I)

= 1 - (19/40) * (5/7)/(5/8)

= 16/35