Application of Bayesian theorem in spam detection & medical diagnosis

Example 1: Spam Detection

Let’s say historically, 20% of emails are spam, so

and the probability that the email is not spam is

.

Suppose the probability of observing the word “free” in a spam email is 0.8, , and in a non-spam email is 0.05, .

Now, if we receive an email containing the word “free”, we want to know the probability that it’s spam.

Using Bayes’ Theorem:

We need to calculate , the overall probability of observing the word “free”:

Now, plugging into Bayes’ Theorem:

So, given that the email contains the word “free”, there’s an 80% chance it’s spam.

Example 2: Medical Diagnosis

Suppose the prevalence of a rare disease is 0.01, so and .

Let’s say the sensitivity of the test (probability of a positive result given the patient has the disease) is 0.95, , and the specificity (probability of a negative result given the patient doesn’t have the disease) is 0.90, .

If a patient tests positive, we want to find the probability they actually have the disease.

Using Bayes’ Theorem:

We need to calculate , the overall probability of testing positive:

Now, plugging into Bayes’ Theorem:

So, if the patient tests positive, there’s approximately an 8.76% chance they actually have the disease.