Question

Suppose 80% of the messages coming into your school’s email system are spam. Because of fairly effective spam filters, suppose only 10% of spam messages are actually delivered to you. Also suppose all non-spam messages are delivered to you.

Let S denote the event that a message is spam, and let D denote the event that a message is delivered to you.

A. What is the probability that a message intended for you is spam and is delivered to you? (Round your answer to 4 decimal places).

You most likely made a tree in the question above about email messages. Transfer those values from the tree into the joint probability table below:

	Message is Spam	Message is Not Spam
Message is Delivered to You	A	C
Message is Not Delivered to You	B	D

Using the information in the previous two questions, answer the following:

b) What is the probability that a message is spam if it is delivered to you? (Round your answer to 4 decimal places).

Answer 1

S denote the event that a message is spam and S' denote the event that a message is not a spam

D denote the event that a message is delivered to you and D' denote the event that a message is not delivered to you.

Tree diagram looks like ,

a) What is the probability that a message intended for you is spam and is delivered to you ?

P( S and D) = P( S)*P( D | S) = 0.8 *0.1

P( S and D) = 0.08

b) What is the probability that a message is spam if it is delivered to you?

P( S | D ) = P( S and D ) / P(D)

First we need to find P(D) = P( D|S)*P(S) + P( D|S')*P(S')  

= (0.8 *0.1) + ( 1*0.2)

= 0.08+0.2

  P(D) = 0.28

P( S | D ) = P( S and D ) / P(D) = 0.08 / 0.28

P( S | D ) = 0.2857