Solved Problems on Prior Probability

Problem 1: Given that 2% of the emails are spam. What is the prior probability of an email being spam?

Solution:

As it is given that 2% of the emails are spam, so the prior probability of an email being spam is given below:

∴ P(Spam) = 2% = 0.02

Problem 2: Consider a medical scenario where the historical data shows that 1% of the total population has the certain disease, then find the prior probability of the patient having certain disease.

Solution:

As it is given that 1% of the total population has the certain disease, so the prior probability of the patient having certain disease is expressed below:

∴ P(Disease) = 1% = 0.01

Problem 3: Imagine the situation where the bank wants to assess the risk of a borrower defaulting on a loan. Historically, 5% of borrowers default on their loans. The bank uses a credit scoring system that correctly predicts defaults with 85% sensitivity (true positive rate) and correctly predicts non-defaults with 90% specificity (true negative rate). If a borrower receives a positive result on this credit scoring system indicating potential default then find what is the probability that the borrower will actually default on the loan?

Solution:

Given,

• Prior probability that a borrower defaults is 5%
• Sensitivity of the credit scoring system is 85% which means that it correctly identifies 85% of those who will default
• Specificity is 90% which means that it correctly identifies 90% of those who will not default

Using these values, we calculate the overall probability of positive result from the credit scoring system:

∴ P(Positive) = P(Positive | Default) . P(Default) + P(Positive | No Default) . P(No\ Default)

Now, substituting the values given, we get

∴ P(Positive) = (0.85 x 0.05) + ((1-0.9) x (1-0.05))

∴ P(Positive) = (0.85 x 0.05) + (0.1 x 0.95)

∴ P(Positive) = 0.0425 + 0.095

∴ P(Positive) = 0.1375

Now, using Bayes Theorem we find the posterior probability

[Tex]\therefore P(Default|Positive) = \frac{P(Positive|Default) \cdot P(Default)}{P(Positive)}[/Tex]

Substituting the values in the above expression

∴P(Default∣Positive) = (0.85×0.05)/0.1375

∴P(Default∣Positive) = 0.0425/0.1375

∴ P(Default | Positive) = 0.3091

Therefore, after a positive result from the credit scoring system, the probability that the borrower will actually default on the loan is approximately 30.91%.

Understanding Prior probability and its study is important as it helps us to combine the new information with the past data to make better decisions and improve accuracy. Prior probability forms as a foundation of Bayesian Theorem which allows us to integrate the new data with the old data to improve the estimation accuracy.

In this article, we will take a deeper look into prior probability, its applications in various fields and some examples for understanding it better.