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Bayesian True Positive Calculator

Stop falling for the base rate fallacy. Calculate the actual probability of a positive result being correct using Bayes' Theorem.

Base Rate (Prior)

%

P(A): The probability of the condition being true *before* the test (e.g., 1% of the population has this disease).

Sensitivity

%

P(B|A): True Positive Rate. Chance the test is positive when the condition is actually true.

Specificity

%

P(~B|~A): True Negative Rate. Chance the test is negative when the condition is actually false.

Running Bayesian inference...

Adjust the sliders to see how the "True Positive" probability changes.

Understanding Bayes' Theorem and True Positives

Bayes' Theorem is a fundamental principle in probability theory that describes how to update the probability for a hypothesis as more evidence or information becomes available. In practical terms, it helps us answer a critical question: "Given a positive test result, what is the actual probability that I have the condition?"

The Base Rate Fallacy

The human brain is notoriously bad at intuitive statistics. Most people, when told a test is 99% accurate and they tested positive, believe there is a 99% chance they have the condition. However, if the condition is very rare (e.g., 1 in 1,000), the actual probability is much lower—often around 9%. This error in judgment is called the Base Rate Fallacy. Our calculator helps you visualize this by factoring in the "Prior Probability" or base rate.

Sensitivity

The probability that the test correctly identifies a positive result (True Positive Rate). High sensitivity means very few "False Negatives."

Specificity

The probability that the test correctly identifies a negative result (True Negative Rate). High specificity means very few "False Positives."

Why Bayesian Logic Matters

  • Medical Screening: Interpreting results for rare diseases where false alarms can lead to unnecessary anxiety or invasive procedures.
  • Cybersecurity: Evaluating intrusion detection systems where a high volume of traffic can lead to thousands of false alerts daily.
  • Legal Evidence: Understanding the weight of forensic evidence, such as DNA matches or eye-witness testimony, in the context of a suspect's prior likelihood of guilt.

Authoritative Statistical Resources

Frequently Asked Questions

  • Can a 100% accurate test be wrong?

    Mathematically, a truly 100% accurate test (100% sensitivity and 100% specificity) would always be correct. However, in the real world, no test is perfect. Even a 99.9% accurate test can fail if the base rate is extremely low.

  • How does Bayes' Theorem apply to AI?

    Modern AI models, especially in classification and medical diagnosis, use Bayesian inference to weigh new data against known probabilities to improve prediction accuracy.

  • What is "Prior" vs "Posterior"?

    The "Prior" is your belief *before* seeing new evidence. The "Posterior" is your updated belief *after* factoring in the new evidence (like a test result).

Note: This tool is intended for educational and illustrative purposes. Always consult with a qualified professional when interpreting real-world medical or technical diagnostic results.