Question

1. A software company is trying to use Bayes Theorem and other rules of probability to develop an algorithm that can effectively filter out spam emails. The company’s software developers carefully examined a random sample of 20,000 emails received by its employees and found out that 4,400 of the emails were spam. A closer look at the spam emails revealed that 1,100 of them contained the word ‘free’ in the subject line. On the other hand, only 780 of the non-spam emails had the word ‘free’ in their subject line. Please answer the following questions based on the given information.

a. What is the probability that a random email will be a spam and will not contain the word ‘free’ in its subject line? Please show your work.            

b. If the word ‘free’ does not appear in the subject line of an email, what is the probability that the email is not a spam? Please show work.                  [2 points]

1. c. What is the probability that a random email will be neither a spam and nor will contain the word ‘free’ in its subject line? Please show your work.         [2 points]                                
   1. d. Are an email being a spam and its subject line containing the word ‘free’ independent events? Please show how you arrived at your answer.      [1 point]

Answer 1

solution:

Given that

Total No.of e-mails = n(S) = 20,000

No.of spam mails = 4,400

   No.of non - spam mails = 20,000 - 4,400 = 15,600

No.of spam mails that contain word 'free' = 1,100

   No.of non-spam mails that contain word 'free' = 780

Let's tabulate them as following:

Type	Free	No-free	Total
Spam	1,100	3,300	4,400
non-spam	780	14,820	15,600
Total	1,880	18,120	20,000

a) Let S = event that randomly selected mail is spam

A = event that randomly selected mail doesn't contain word 'free'

P(A) = n(A) / n(S) = 18,120/ 20,000 = 0.906

P( random email will be spam and doesn't contain word 'free' ) = P(A S)

= n(A S) /n(S)

= 3,300/20,000

= 0.165

Probability that randomly selected email will be spam and doesn't contain word 'free' = 0.165

b) Let N= event that randomly selected mail is non-spam

P(randomly selected mail is non-spam given that selected mail doesn't contain word 'free') = P( N| A)

= P(A N) /P(A)

= (14,820 / 20,000) / 0.906

= 0.818

Probability that randomly selected email will be non-spam and doesn't contain word 'free' = 0.818

c) Let F = event that randomly selected mail contain word 'free'

P(F) = n(F) /n(S) = 1,880/20,000 = 0.094

S F = event either spam or contain word 'free'

( S F ) ' = event neither spam nor contain word 'free'

P( S F) = P(S) + P(F) - P(S F) = 4,400 / 20,000 + 1,880 / 20,000 - 1,100/20,000

= 5180 / 20,000

= 0.26

Now, P ( S F )' = 1 -  P( S F) = 0.74

Probability that randomly selected mail is neither spam nor contain word 'free' = 0.74

d) We need to check events S and F are independent or not.

If S and F are independent then P(S F) = P(S) * P(F)

we have P(S F) = 0.055

P(S) * P(F) = 0.22 *0.094

= 0.02

!= P(S F)

    S and F are not Independent events

   An email being a spam and its subject line containing the word ‘free’ are not independent events