Question

Problem 5. (12 points) Bayes' Theorem is a popular tool for spam filtering. You are asked to design a spam filtering algorithm based on whether certain words appear in an email. You were given 1,000 randomly selected emails that entered an email server. You examined these emails and manually labeled each one either as spam or ham (i.e., non-spam). You found that 400 emails are spam and 600 are ham. In these 400 spam emails, you found 200 of thenm include the word "password; in the 600 ham emails, you found 30 include the word "password". Now use the Bayes' Theorem to answer the following two questions: (1) If there's a new email entering the server and it includes the word "password", what is the probability that it is a spam? (2) If there's a new email entering the server and it does NOT include the word "password", what is the probability that it is a spam?

Answer 1

Let S shows the event that email is spam and H shows the event that email is ham. Let P shows the event that email include password. So we have

P(S) = 400 /1000 = 0.40, P(H) = 600/1000 = 0.60

and

P(P|S) = 200 /400 = 0.50

P(P|H) = 30 / 600 = 0.05

Let N shows the event that email does not contain password. So,

P(N|S) = 1- P(P|S) = 1 - 0.50 = 0.50

P(N|H) = 1 - P(P|H) = 1- 0.05 = 0.95

(1)

Using Baye's theorem the required probability is

P(S|P) = [P(P|S)P(S)]/ [ P(P|S)P(S)+P(P|H)P(H)] = [0.50 * 0.40 ] / [ 0.50*0.40 + 0.05 * 0.60] = 0.8696

(2)

Using Baye's theorem the required probability is

P(S|N) = [P(N|S)P(S)]/ [ P(N|S)P(S)+P(N|H)P(H)] = [0.50 * 0.40 ] / [ 0.50*0.40 + 0.95 * 0.60] = 0.2597