Bayesian Inference Calculator

Bayesian Inference Calculator: run bayesian inference calculations online. Formula, assumptions, and interpretation guide.

A probability calculator computes the likelihood of events occurring, including simple probabilities, combined events (AND/OR), conditional probabilities (Bayes' theorem), permutations, and combinations. Probability underpins statistics, gambling, insurance, genetics, quality control, and virtually every area of quantitative decision-making. For Bayesian questions, continue with the , prior-posterior calculator, and Bayes factor calculator.

Understanding probability helps you evaluate risk accurately, avoid gambler's fallacy, assess medical test results, and make better decisions under uncertainty in everyday and professional contexts.

Choose the calculation type: simple probability, combined events, conditional probability, permutations, or combinations.
Enter the number of favourable outcomes and total possible outcomes for simple probability.
For AND (both events): multiply individual probabilities (if independent).
For OR (at least one event): use P(A∪B) = P(A) + P(B) − P(A∩B).
For combinations (nCr) or permutations (nPr), enter n (total items) and r (items chosen).

Key probability formulas

Simple probability: P(A) = favourable outcomes / total outcomes

AND (independent): P(A ∩ B) = P(A) × P(B)

OR: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Conditional: P(A|B) = P(A ∩ B) / P(B)

Combinations: nCr = n! / (r! × (n−r)!)

Permutations: nPr = n! / (n−r)!

Interpreting probability results

Converting between probability formats

Probability 0.25 = 25% = 1 in 4 = odds of 1:3. To convert probability p to odds: odds = p / (1−p). Odds of 3:1 against means probability = 1/4 = 0.25. Expected value = Σ(outcome × probability) — a positive expected value means the bet or decision is profitable in the long run, negative means you expect to lose on average.

Statistics tips and best practices

Independent vs dependent events: drawing cards without replacement changes probabilities with each draw — account for this.
Bayes' theorem is essential for interpreting medical test results — a positive result for a rare disease with a 99% accurate test may still be more likely false than true due to the low base rate.
The gambler's fallacy: past independent outcomes do not affect future ones — a coin has no memory.
Complement rule: P(A not occurring) = 1 − P(A) — often easier to compute than the event itself.

The probability of winning the US Powerball jackpot is approximately 1 in 292 million (about 0.00000034%).
A medical test with 99% sensitivity and 99% specificity applied to a disease affecting 1 in 10,000 people: a positive result is still more likely to be a false positive than a true positive (Bayesian base rate effect).
The probability that at least two people in a room of 23 share a birthday is approximately 50% — the famous "birthday paradox."

Common mistakes to avoid

Assuming independence when events are not — drawing cards, sampling without replacement, and correlated events all violate independence.
Confusing P(A|B) and P(B|A) — the probability that a test is positive given disease ≠ the probability of disease given a positive test.
Adding probabilities of non-mutually-exclusive events without subtracting the overlap — double-counting the intersection.

Probability calculations are mathematical tools for modelling uncertainty. Gambling decisions should account for the house edge and expected losses. Medical probability interpretations should involve a qualified clinician.