1. What is Bayes' Theorem?

Bayes' Theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event. It's fundamental in probability theory and statistics, particularly in updating beliefs with new evidence.

2. How Does the Calculator Work?

The calculator uses Bayes' Theorem:

Where:

• \( P(A|B) \) — Posterior probability (probability of A given B)
• \( P(B|A) \) — Likelihood (probability of B given A)
• \( P(A) \) — Prior probability of A
• \( P(B) \) — Marginal probability of B

Explanation: The theorem updates our initial belief (prior) about event A after observing event B.

3. Application to Coin Flips

Example: If you suspect a coin might be biased (prior), Bayes' theorem lets you update this belief after observing a series of flips (evidence).

4. Using the Calculator

Tips: Enter probabilities between 0 and 1. For coin flips, P(A) might be your initial belief about the coin being fair (e.g., 0.5), P(B|A) is the probability of observing your flip results given a fair coin, and P(B) is the total probability of observing your flip results.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from regular probability?
A: Bayes' Theorem incorporates prior knowledge or beliefs, updating them with new evidence.

Q2: Can I use this for more than two outcomes?
A: Yes, Bayes' Theorem can be extended to multiple hypotheses and outcomes.

Q3: What's a practical coin flip example?
A: If you get 7 heads in 10 flips, Bayes' Theorem helps calculate how likely the coin is biased given your initial belief.

Q4: Why is the marginal probability important?
A: P(B) normalizes the result, ensuring probabilities sum to 1 across all possibilities.

Q5: Can I use this for continuous distributions?
A: Yes, there's a continuous version of Bayes' Theorem used in more advanced applications.