Question

You are designing a new cloud-based, AI-powered suite of security software for home users. You are currently working on the module to detect phishing and other malicious e-mails. Based on extensive analysis of millions of e-mails you gathered using a bogus Facebook app, you have determined the following:

i. 68% of malicious emails contain the phrase “weird trick”

ii. 3% of valid email messages contain the phrase “weird trick”

iii. 22% of all email messages are malicious

Determine the following probabilities:

a). A message contains the phrase “weird trick”

b). The message is malicious, given that it contains the phrase “weird trick”

c). A message is not malicious, given that it does not contain the phrase “weird trick”

Answer 1

Let the event of getting malicious emails be A, getting valid

emails be B and getting the phrase "weird trick" be C.

P(A) = 0.22, P(B) = 1 - 0.22 = 0.78, P(C | A) = 0.68,

P(C | B) = 0.03

a. The probability that a message contains the phrase "weird

trick" = P(C) = P(C | A) P(A) + P(C | B) P(B)

= (0.68 * 0.22) + (0.03 * 0.78) = 0.173. (Ans).

b. The probability that the message is malicious given that it

contains the phrase "weird trick" = P(A | C) = P(A C) / P(C)

= [P(C | A) P(A)] / P(C) = (0.68 * 0.22) / 0.173 = 0.8647. (Ans).

c. The probability that a message is not malicious given that

it does not contain the phrase "weird trick" = P( | )

= P(B | ) [As, = B] = P(B) - P(B | C)

[As, P(B) = P(B | C) + P(B | )]

P(B | C) = [P(C | B) P(B)] / P(C) = (0.03 * 0.78) / 0.173 = 0.1353.

The required probability = 0.78 - 0.1353 = 0.6447. (Ans).