Bayes' Theorem Calculator
Calculate posterior probability using Bayes' Theorem
Input Probabilities
Enter probabilities as percentages (0-100)
Initial probability that event A occurs
Probability of B occurring given that A is true
Probability of B occurring given that A is false
Load Example
Bayes' Theorem Formula
Posterior Probability
P(A|B) - Probability of A given B
All Probabilities
Calculation Steps
Interpretation
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About Bayes' Theorem Calculator
What is Bayes' Theorem?
Bayes' Theorem is a fundamental formula in probability theory that describes how to update the probability of a hypothesis based on new evidence. It allows us to revise our beliefs (prior probabilities) when we receive new information (likelihood), resulting in updated beliefs (posterior probabilities).
The Formula
P(A|B) = [P(B|A) × P(A)] / P(B)
- P(A|B) - Posterior: Probability of A given evidence B
- P(B|A) - Likelihood: Probability of observing B if A is true
- P(A) - Prior: Initial probability of A before seeing evidence
- P(B) - Marginal: Total probability of observing B
Real-World Applications
- Medical Diagnosis: Calculating disease probability from test results
- Spam Filtering: Determining if an email is spam based on word patterns
- Machine Learning: Naive Bayes classifiers for text and data classification
- Weather Forecasting: Updating predictions based on new observations
- Legal Reasoning: Evaluating evidence in court cases
- Finance: Risk assessment and portfolio optimization
- Search Engines: Ranking and relevance algorithms
- Robotics: Sensor fusion and localization
Medical Test Example
A classic example: A disease affects 1% of the population. A test is 99% accurate for those with the disease (true positive) but has a 5% false positive rate. If you test positive, what's the actual probability you have the disease? Surprisingly, it's only about 16.5%! This counterintuitive result shows why Bayes' Theorem is crucial for interpreting medical tests.
Key Insights
- The posterior probability depends on both the prior and the likelihood
- Strong evidence can overcome weak priors, and vice versa
- Bayes' Theorem is the foundation of Bayesian statistics
- It provides a mathematical framework for rational belief updating
- The theorem is reversible: you can calculate P(B|A) from P(A|B)
Common Misconceptions
- P(A|B) ≠ P(B|A) - These are different conditional probabilities
- A high accuracy test doesn't guarantee a high probability of disease
- Base rates (priors) matter significantly in real-world applications
- Ignoring P(B) leads to the "base rate fallacy"
History
Named after Reverend Thomas Bayes (1701-1761), the theorem was published posthumously in 1763. It has become one of the most important tools in statistics, artificial intelligence, and scientific reasoning.