Calculate single, multiple, and conditional probabilities, plus permutations (nPr) and combinations (nCr) — with fractions, percentages, odds, and Bayes theorem.
A full-featured probability toolkit for students, statisticians, and decision-makers. Compute simple event probabilities, combine independent or dependent events, apply Bayes theorem for posterior probabilities, and evaluate combinations and permutations with exact BigInt math up to n=1000. Every result is shown as a fraction, decimal, percentage, odds ("1 in X"), and a 100-dot visual grid. Pre-built examples cover coin flips, dice, lotteries, card draws, and medical-test Bayesian inference.
Probability sits between 0 (impossible) and 1 (certain). For a single event with equally likely outcomes, P(A) equals favorable outcomes divided by total outcomes. When combining events, the rule depends on their relationship: independent events (two coin flips) multiply — P(A and B) = P(A) × P(B) — while mutually exclusive events (rolling a 1 OR a 2 on one die) add — P(A or B) = P(A) + P(B). For inclusive OR with independent events, subtract the overlap to avoid double-counting.
Conditional probability P(A|B) — read "the probability of A given B" — answers questions like "given that a test came back positive, how likely is the disease?". Bayes theorem flips the conditioning: P(A|B) = [P(B|A) × P(A)] / P(B), where P(B) is the total probability of the evidence. This is why a 95%-accurate test for a rare disease (1% prevalence) can still produce a posterior probability under 20% — the false-positive rate dominates when the prior is small.
Use permutations (nPr = n! / (n-r)!) when the arrangement counts: gold/silver/bronze medals, password orderings, seating plans. Use combinations (nCr = n! / (r! × (n-r)!)) when the selection alone matters: lottery picks, poker hands, committee membership. A handy mental check — if swapping two chosen items produces a different outcome, you want permutations; if they are the same, use combinations.
Permutations count arrangements where order matters (1st, 2nd, 3rd place finishers); combinations count selections where order does not (a 5-card hand is the same hand regardless of deal order). nPr is always greater than or equal to nCr.
Use Bayes whenever you need to update a prior belief P(A) after observing evidence B. Classic uses: medical-test accuracy, spam filters, fault diagnosis. The calculator derives P(A|B) from your prior, the true-positive rate P(B|A), and the false-positive rate.
Yes. The calculator uses JavaScript BigInt to avoid floating-point overflow, so values like C(52,5) or C(49,6) are computed exactly. n is capped at 1000 for performance.
It means the event is expected to occur once for every 6 equally likely trials, on average. It is just 1 ÷ probability — useful for intuition (a die roll giving a 6 is 1 in 6).
Mutually exclusive events cannot both happen (rolling a 1 OR a 2). Independent events can both happen and do not influence each other (two coin flips). For exclusive OR you simply add; for independent OR you add then subtract the overlap.
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