Question

It is estimated that 50% of emails are spam emails. Some software has been
applied to filter these spam emails before they reach your inbox. A certain brand of software
claims that it can detect 99% of spam emails, and the probability for a false positive (a
non-spam email detected as spam) is 5%.
Now if an email is detected as spam, then what is the probability that it is in fact a non spam email ?

Answer 1

Let AA denote the event that an email is detected as spam and BB denote the event that an email is spam.

Given that 50% of the emails are spam, i.e., P(B)=0.5P(B)=0.5. Thus P(B′)=1−P(B)=0.50P(B′)=1−P(B)=0.50.

A certain brand of software claims that it can detect 99% of spam emails. That is P(A|B)=0.99P(A|B)=0.99.

And the probability for a false positive (a non-spam email detected as spam) is 5%. That is P(A|B′)=0.05P(A|B′)=0.05.

We need to find the probability that the email is non-spam given that it is detected as spam.

Using Bayes' Theorem, required probability is

P(B′|A)