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Bayes' Theorem Calculator - Online Probability Update Tool

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Bayes' Theorem Calculator

Update probabilities with new evidence ā€“ compute posterior probability instantly

Input Probabilities
prob
Please enter a value between 0 and 1 (or 0ā€“100 in % mode).
Your belief before seeing evidence (e.g., disease prevalence)
prob
Please enter a value between 0 and 1 (or 0ā€“100 in % mode).
Probability of the evidence given the hypothesis is true (sensitivity)
prob
Please enter a value between 0 and 1 (or 0ā€“100 in % mode).
Probability of the evidence when the hypothesis is false (1 ā€“ specificity)
Updated Belief
16.1%

Posterior Probability P(A|B)

(approximately 0.161)

Bayes' Formula:
P(A|B) = P(B|A) Ā· P(A) / [ P(B|A) Ā· P(A) + P(B|Ā¬A) Ā· (1 āˆ’ P(A)) ]
Step-by-step calculation:
  • Numerator: 0.95 Ɨ 0.01 = 0.0095
  • Denominator: 0.0095 + 0.05 Ɨ 0.99 = 0.0095 + 0.0495 = 0.0590
  • Posterior: 0.0095 / 0.0590 ā‰ˆ 0.1610 (16.1%)
Even with a highly accurate test (95% sensitivity), if the condition is rare (1% prevalence) and false positives exist (5%), the probability of actually having the condition after a positive result is only about 16.1%.

Frequently Asked Questions

Bayes' Theorem describes the probability of an event based on prior knowledge of conditions related to the event. It's mathematically expressed as P(A|B) = [P(B|A) * P(A)] / P(B). It's a fundamental concept in probability theory and statistics, widely used in medical testing, machine learning, and risk analysis.

The posterior probability P(A|B) is your updated belief about event A after observing evidence B. For example, if the posterior is 16%, it means that given a positive test result, there is only a 16% chance the person actually has the condition (if the condition is rare and false positives exist).

Prior probability P(A) is your initial estimate of how likely an event is before considering new evidence. In medical contexts, it's often the prevalence of a disease in the general population.

The false positive rate P(B|Ā¬A) is the probability of observing the evidence even when the hypothesis is false. For a medical test, it's the chance of testing positive when you do not have the disease (1 minus specificity).

Yes, toggle the ā€œUse percentagesā€ switch to enter values between 0 and 100. The calculator will automatically convert them to decimals internally. The output is shown both as a percentage and a decimal.

Bayes' theorem helps make rational decisions under uncertainty by combining prior knowledge with new data. It's used in spam filters, medical diagnosis, courtrooms (probability of guilt given DNA evidence), and even in A/B testing for websites.
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