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:

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

Explanation: The theorem updates our belief about A after seeing B.

3. Application to Coin Flips

Details: For coin flips, we might use Bayes' theorem to update our belief about whether a coin is fair after observing a series of heads and tails.

4. Using the Calculator

Tips: Enter probabilities between 0 and 1. For coin flip examples, prior might be 0.5 (fair coin), likelihood depends on your hypothesis, and marginal probability is the total probability of the observed data.

5. Frequently Asked Questions (FAQ)

Q1: What's a practical coin flip example?
A: If you get 7 heads in 10 flips, Bayes' theorem helps calculate the probability the coin is biased.

Q2: Why use Bayes' theorem for coin flips?
A: It provides a mathematical way to update beliefs about fairness as you see more data.

Q3: What's the difference between prior and posterior?
A: Prior is your initial belief, posterior is your updated belief after seeing evidence.

Q4: How do I determine the marginal probability?
A: For coin flips, it's the probability of the observed data under all possible hypotheses.

Q5: Can this be used for other probability problems?
A: Yes, Bayes' theorem is widely applicable in statistics, machine learning, and data science.