Calculate probabilities across 4 modes: single events, combined AND/OR events, conditional probability (Bayes), and combinations & permutations (nCr/nPr). Shows step-by-step working, Venn diagram, and a probability scale visualisation.
Single Event: enter favourable outcomes and total outcomes. Combined: enter P(A) and P(B) to get P(A∩B), P(A∪B), P(A|B). Conditional: apply Bayes' theorem with prior probability, sensitivity, and specificity. Combinations/Permutations: calculate nCr (choose r from n) or nPr (arrange r from n).
Fill in the required fields — the tool validates that probabilities are between 0 and 1. Odds are converted to probabilities automatically if entered as X:Y format. The result shows the probability as a decimal, percentage, fraction, and on a 0–1 probability scale bar.
A Venn diagram canvas illustrates event overlaps for combined events. Combinations mode shows Pascal's triangle highlighting the result. The step-by-step working traces every formula and substitution so you can follow the calculation by hand.
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Classical probability: P(event) = (number of favourable outcomes) / (total equally likely outcomes). Example: rolling a 4 on a fair die: P = 1/6 ≈ 0.1667. Frequentist probability estimates likelihood from observed frequency: 47 heads in 100 flips → P(head) ≈ 0.47. Probabilities can be expressed as decimals (0.25), percentages (25%), fractions (1/4), or odds (1:3, meaning 1 success for every 3 failures).
P(A and B) = P(A∩B) is the probability that BOTH events occur simultaneously. For independent events: P(A∩B) = P(A)×P(B). For dependent events: P(A∩B) = P(A)×P(B|A). Example: P(drawing two aces from a deck without replacement) = 4/52 × 3/51 = 12/2652 ≈ 0.0045. P(A or B) = P(A∪B) is the probability that AT LEAST ONE event occurs: P(A∪B) = P(A) + P(B) − P(A∩B). The subtraction avoids double-counting the overlap. For mutually exclusive events (P(A∩B)=0): P(A∪B) = P(A) + P(B).
Conditional probability P(A|B) is the probability of A given that B has already occurred: P(A|B) = P(A∩B)/P(B). Example: P(drawing a queen | drawing a face card) = P(queen AND face card)/P(face card) = (4/52)/(12/52) = 4/12 = 1/3. Bayes' theorem reverses conditional probability: P(B|A) = P(A|B)×P(B)/P(A). It is fundamental in medical diagnosis (sensitivity/specificity), machine learning (Naive Bayes classifiers), spam filters, and A/B testing.
Both count ways to select items from a set. Permutations (nPr): order MATTERS. Number of ways to arrange r items from n: nPr = n!/(n−r)!. Example: seating 3 people from 5 in 3 chairs: 5P3 = 5×4×3 = 60. Combinations (nCr): order does NOT matter. Number of ways to choose r items from n: nCr = n!/(r!(n−r)!). Example: choosing 3 from 5 for a committee (order irrelevant): 5C3 = 10. nPr = nCr × r! because each combination generates r! different arrangements. Pascal's triangle encodes all binomial coefficients C(n,r).
Bayes' theorem: P(H|E) = P(E|H)×P(H)/P(E), where H is a hypothesis and E is observed evidence. It updates prior probability P(H) based on new evidence E to give posterior probability P(H|E). Example: a medical test for a disease with 1% prevalence, 95% sensitivity (true positive rate), 90% specificity (true negative rate). If you test positive: P(disease|positive) = (0.95×0.01)/(0.95×0.01 + 0.10×0.99) ≈ 8.7%. Despite a "95% accurate" test, a positive result only means ~9% chance of disease when prevalence is low — a counterintuitive but critical insight for medical decision-making.
The Law of Large Numbers states that as the number of trials increases, the experimental probability converges to the theoretical probability. Flipping a fair coin 10 times might give 7 heads (70%), but after 10,000 flips the proportion will be very close to 50%. The strong version guarantees convergence with probability 1. The law underlies insurance (pooling many independent risks makes outcomes predictable), casino profitability (house edge over millions of bets), and statistical sampling (large samples are representative). It does NOT mean that past outcomes influence future ones — the "gambler's fallacy" is the incorrect belief that a run of heads makes tails more likely.
