Question

Email Subject Word		Spam	Frequency
‘Free’	‘Sale’
Y	Y	Y	20
Y	Y	N	8
Y	N	Y	6
Y	N	N	11
N	Y	Y	4
N	Y	N	19
N	N	Y	10
N	N	N	37
Total			115

We wish to create a spam filter. So as a first pass at this task we wish to determine some association rules based on the sample set of email data given in the above table. Based on the above table (provide all answers to 3 significant digits):

1. What is the probability that an email has both the word ‘Free’ in its subject line and is spam?
2. What is the probability that an email with the word ‘Free’ in its subject line is spam?
3. What is the probability that an email without the word ‘Free’ in its subject line is spam?
4. What is the probability that an email with the word ‘Sale’ in its subject line is spam?
5. What is the probability that an email without the word ‘Sale’ in its subject line is spam?
6. (1 mark) What single property (e.g. the word ‘Free’ is in the subject line) is the most likely to indicate that an email is spam (hint: use association rules)?
7. What complete set of properties (i.e. assigning values over all properties of ‘Free’ and ‘Sale’) is most likely to indicate that an email is spam?
8. (1 mark) What property (i.e. over all possible single or combination) is the most likely to indicate that an email is spam?

Answer 1

(a) The probability that an email has both the word 'Free' in its subject line and is spam = P('Free' = Y, Spam = Y) = (20 + 6)/115 = 0.2261

(b) The required probability = P(Spam = Y | 'Free' = Y)

= (20 + 6)/(20 + 8 + 6 + 11) = 26/45 = 0.5778

(c) The required probability = P(Spam = Y | 'Free' = N)

= (4 + 10)/(4 + 19 + 10 + 37) = 14/70 = 0.20

(d)The required probability = P(Spam = Y | 'Sale' = Y)

= (20 + 4)/(20 + 8 + 4 + 19) = 24/51 = 0.4706

(e) The required probability = P(Spam = Y | 'Sale' = N)

= (6 + 10)/(6 + 11 + 10 + 37) = 16/64 = 0.25

(f) The word 'Free' in the subject line is the most likely property to indicate that an email is spam

(g) Both the words 'Free' and 'Sale' in the subject line is the most likely complete set of properties which indicate that an email is spam

(h) The word 'Free' in the subject line is the most likely property to indicate that an email is spam