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 5,000 emails received by its employees and found out that 1200 of the emails were spam. A closer look at the spam emails revealed that 300 of them contained the word ‘free’ in the subject line. On the other hand, only 228 of the non-spam emails had the word ‘free’ in their subject line. Please answer the following questions based on the given information
2. 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.                           [2 points
3. 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]
4. 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]          
5. 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

From the given information: The following table is formulated:

	Contain the word free	Does not contain the word free	Total
Spam	300		1200
not Spam	228
Total			5000

And the blank cell values are filled:

	Contain the word free	Does not contain the word free	Total
Spam	300	1200-300=900	1200
not Spam	228	3800-228=3572	5000-1200=3800
Total	528	3572+900=4472	5000

Probability that a random email will be a spam and will not contain the word ‘free’ in its subject line

= Number of spam emails that does not contain the word free in the subject line/Total number of emails

From the above table,

Number of spam emails that does not contain the word free in the subject line = 900

Total number of mails = 5000

Probability that a random email will be a spam and will not contain the word ‘free’ in its subject line

= Number of spam emails that does not contain the word free in the subject line/Total number of emails = 900/5000=0.18

Probability that a random email will be a spam and will not contain the word ‘free’ in its subject line = 0.18

If the word ‘free’ does not appear in the subject line of an email, probability that the email is not a spam =

Number of emails that does not contain word "free" in the subject line and not a spam / Number of emails that does not contain word "free" in the subject line

From the above table,

Number of not spam emails that does not contain the word free in the subject line = 3572

Number of emails that does not contain word "free" in the subject line = 4472

If the word ‘free’ does not appear in the subject line of an email, probability that the email is not a spam

= Number of emails that does not contain word "free" in the subject line and not a spam / Number of emails that does not contain word "free" in the subject line

= 3572/4472 = 0.798747764

If the word ‘free’ does not appear in the subject line of an email, probability that the email is not a spam = 0.798747764

Probability that a random email will be neither a spam and nor will contain the word ‘free’ in its subject line

= Number of emails that does not contain word "free" in the subject line and not a spam / Total number of emails

From the above table,

Number of not spam emails that does not contain the word free in the subject line = 3572

Total number of emails = 5000

Probability that a random email will be neither a spam and nor will contain the word ‘free’ in its subject line

= Number of emails that does not contain word "free" in the subject line and not a spam / Total number of emails

= 3572/5000=0.7144

Probability that a random email will be neither a spam and nor will contain the word ‘free’ in its subject line = 0.7144

Are an email being a spam and its subject line containing the word ‘free’ independent events

Answer : No.

Probability that a email is spam = Number of spam email / total number of mails = 1200/5000=0.24

Probability that an email contains the word " free" in the subject line = Number of emails that contain "free" in the subject line / total number of mails = 528/5000=0.1056

If the email being a spam and its subject line containing the word ‘free’ are independent events then;

Probability that a email is spam and contains the word " free" in the subject line = Probability that a email is spam x Probability that an email contains the word " free" in the subject line

Probability that a email is spam x Probability that an email contains the word " free" in the subject line = 0.24 x 0.1056 = 0.025344

Probability that a email is spam and contains the word " free" in the subject line = Number of spam emails that contain the word "free" in the subject line / Total number of emails = 300/5000 = 0.06

As 0.06 is not equal 0.025344 i.e

Probability that a email is spam and contains the word " free" in the subject line Probability that a email is spam x Probability that an email contains the word " free" in the subject line

Therefore,

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