1. What is Bayesian Rule?

Definition: Bayes' Theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event.

Purpose: It provides a way to update probabilities when given new evidence, widely used in statistics, machine learning, and data science.

2. How Does Bayesian Rule Work?

The formula is:

Where:

• \( P(A|B) \) — Posterior probability: Probability of A given B is true
• \( P(B|A) \) — Likelihood: Probability of B given A is true
• \( P(A) \) — Prior probability: Initial probability of A
• \( P(B) \) — Marginal probability: Total probability of B

Explanation: The theorem updates our belief about A after seeing B by combining our prior knowledge with the new evidence.

3. Importance of Bayesian Rule

Details: Bayesian reasoning is fundamental in many fields including medical testing, spam filtering, and machine learning algorithms.

4. Using the Calculator

Tips: Enter probabilities between 0 and 1. P(B) must be greater than 0. All inputs represent probabilities and should be ≤ 1.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between P(A|B) and P(B|A)?
A: P(A|B) is the probability of A given B occurred, while P(B|A) is the probability of B given A occurred - they're not the same!

Q2: How do I calculate P(B)?
A: P(B) = P(B|A)P(A) + P(B|¬A)P(¬A) - the total probability of B occurring.

Q3: Can probabilities be greater than 1?
A: No, all probabilities must be between 0 and 1. If your result is >1, check your inputs.

Q4: What's a practical example of Bayes' Theorem?
A: Medical tests - determining the probability of having a disease given a positive test result.

Q5: Why is it called "Bayesian"?
A: Named after Reverend Thomas Bayes who formulated the first version of this probability theory.