Question

Problem 1 (Bayes theorem and spam filters) Suppose you have develop a new algorithm to detect spam in an incoming email message. if the email is spam, there is a 98% chance your algorithm will detect it. On the other hand, if no spam is present, there is a 90% chance the algorithm will indicate that the message is not spam. Suppose that roughly 10% of all your email is spam. a) What is the probability a randomly chosen message coming into your email will be detected as being "spam" with your new algorithm? b) What is the probability that if a message is flagged as spam by your algorithm that it actually is? c) What is the probability that if a message is flagged as spam by your algorithm that it actually is not? d) What is the probability that in 100 incoming email messages, your algorithm will misclassify (iue it will let a message that is spam get through) at least one spam message? (Hint: in 100 email messages, how many, on average, will be spam?)

Answer 1

Let the event that an email is spam be denoted as S $\Rightarrow$ The event that an email is not spam is denoted as S^c   

Let the event that an email is detected as spam be denoted as D $\Rightarrow$ The event that an email is not detected as spam is denoted as D^c

Given:- P(S)=0.1 $\Rightarrow$ P(S^c)=1-0.1=0.9

   P(D|S)=0.98 $\Rightarrow$ P(D^c|S)=1-0.98=0.02  

   P(D^c|S^c)=0.9 $\Rightarrow$ P(D|S^c)=1-0.9=0.1