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.

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 prior beliefs about an event (P(A)) with new evidence (P(B|A)) to give a revised probability (P(A|B)).

3. Application to Coin Flips

Details: For coin flips, Bayes' theorem can help determine the probability that a coin is fair given observed flips, or the probability of getting heads given certain prior information.

4. Using the Calculator

Tips: Enter probabilities between 0 and 1. For coin flips, a fair prior would be 0.5. The marginal probability P(B) must be greater than 0.

5. Frequently Asked Questions (FAQ)

Q1: What's a practical example for coin flips?
A: If you get 7 heads in 10 flips, Bayes' theorem can calculate the probability the coin is biased, given your prior belief about its fairness.

Q2: How is this different from frequentist probability?
A: Bayesian probability incorporates prior beliefs, while frequentist probability only considers the observed data.

Q3: What should I use for P(B)?
A: P(B) is often calculated as P(B|A)P(A) + P(B|¬A)P(¬A), the total probability of observing B under all possibilities.

Q4: Can this be used for multiple coin flips?
A: Yes, by updating the posterior sequentially after each flip (Bayesian updating).

Q5: What's a common mistake when using Bayes' theorem?
A: Confusing P(A|B) with P(B|A) - these are different quantities (prosecutor's fallacy).