spam_email_probability

Probability of Actual Spam

Consider an email classification scenario where emails can be categorized into two classes: positive (spam) and negative (non-spam). Among all received emails, 10% belong to the positive class (spam), and the remaining 90% belong to the negative class (non-spam). John's email filter, designed to identify spam, has an accuracy of 90% in correctly classifying positive instances (true positives) but incorrectly labels 10% of negative instances as positive (false positives). If an email is classified as positive by John's filter, what is the probability that it actually belongs to the positive class (spam) ?

Solution

To determine the probability that an email marked as spam by spam filter is actually spam, we can use Bayes' theorem. Let's define the following events:

A: Email is spam
B: Email is marked as spam by the filter

We are asked to find P(A|B), the probability that an email is spam given that it's marked as spam. Bayes' theorem states: \[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \] Where:
\( P(B|A) \) is the probability that the filter marks an email as spam given that it is actually spam.
\( P(A) \) is the probability that an email is spam (10% or 0.1).
\( P(B) \) is the probability that an email is marked as spam.
\( P(B|A) \) is the reliability of the filter, which is 90% or 0.9.
\( P(B) \) can be calculated using the law of total probability:
\[ P(B) = P(B|A) \times P(A) + P(B|\neg A) \times P(\neg A) \]
Where:
\( P(\neg A) \) is the probability that an email is not spam (90% or 0.9).
\( P(B|\neg A) \) is the probability that the filter marks a non-spam email as spam, which is the false positive rate.
Given that \( P(\neg A) = 1 - P(A) = 1 - 0.1 = 0.9 \), and the false positive rate is also 10% or 0.1, we can calculate \( P(B) \):
\[ P(B) = 0.9 \times 0.1 + 0.1 \times 0.9 = 0.09 + 0.09 = 0.18 \] Now we can plug all these values into Bayes' theorem: \[ P(A|B) = \frac{0.9 \times 0.1}{0.18} \] \[ P(A|B) = \frac{0.09}{0.18} \] \[ P(A|B) = 0.5 \] So, the probability that an email marked as spam by spam filter is actually spam is 50%.