Question

According to IT department of leading financial firm, 50% of emails received by their employees are spam. They are planning to install a software which filters the spam emails. A reputable cybersecurity brand has launched a software which claims to detect 99% of spam emails. However, they have also reported 5% as the chance of detecting a nonspam email as spam. If an email is detected as spam, what is the probability of it in fact being a non-spam email? 

Answer 1

Let the event A is that the email is detected as spam,

B₁ - the email is a spam email,

B₂ - the email is not a spam email.

50% of emails are spam,

$P(B_1) =P(B_2)=0.5,\\ P(A|B_1)=0.99,P(A|B_2)=0.05.$

The sought probability we'll find using Bayes theorem:

$P(B_2|A)=\\ =\cfrac{P(A|B_2)\cdot P(B_2)}{P(A|B_1)\cdot P(B_1)+P(A|B_2)\cdot P(B_2)}=\\ =\cfrac{0.05\cdot0.5} {0.99\cdot0.5+0.05\cdot0.5} =\cfrac{0.025}{0.52} =\cfrac{5}{104}=0.0481.$