Question

Bayes' rule can be used to identify and filter spam emails and text messages. This question refers to a large collection of real SMS text messages from participating cellphone users.1 In this collection, 747 of the 5574 total messages () are identified as spam. The word "text" (or "txt") is contained in of all messages, and in of all spam messages. What is the probability that a message is spam, given that it contains the word "text" (or "txt")? Round your answer to three decimal places.

Answer 1

Answer:

0.134 = 13.4% probability that a message is spam, given that it contains the word "text" (or "txt")

Step-by-step explanation:

We use the conditional probability formula to solve this question. It is

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

In which

P(B|A) is the probability of event B happening, given that A happened.

$P(A \cap B)$ is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Contains the word "text", or "txt".

Event B: Spam message.

The word "text" (or "txt") is contained in of all messages, and in of all spam messages.

This means that $P(A) = 1$

747 of the 5574 total messages () are identified as spam. The word "text" (or "txt") is contained in all of them. So

$P(A \cap B) = \frac{747}{5574} = 0.134$

Then

$P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.134}{1} = 0.134$

0.134 = 13.4% probability that a message is spam, given that it contains the word "text" (or "txt")