Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain), often expressed as a percentage. The basic formula is: P(Event) = Number of favorable outcomes / Total number of possible outcomes. For example, rolling a 6 on a fair six-sided die has a probability of 1/6 ≈ 16.67%. This tool computes this instantly and also handles multi-trial scenarios like "at least once" probability.

Permutations (nPr) count arrangements where order matters — for example, the sequence ABC is different from CBA. The formula is n!/(n−r)!. Combinations (nCr) count selections where order doesn't matter — ABC is the same group as CBA. The formula is n!/(r!(n−r)!). Use permutations for races, rankings, or PIN codes; use combinations for lottery picks, team selections, or hand draws.

Decimal odds (common in Europe): Probability = 1 / Decimal Odds. E.g., odds of 4.0 imply a 25% chance.
Fractional odds (common in UK): a/b means probability = b/(a+b). E.g., 3/1 implies 25%.
American odds: Positive (+300) → probability = 100/(odds+100) = 25%. Negative (-200) → probability = |odds|/(|odds|+100) ≈ 66.7%.
Our Odds Converter tab handles all three formats instantly.

When you repeat an independent event n times, the probability that it occurs at least once is: 1 − (1−p)ⁿ, where p is the single-trial probability. For instance, if you roll a die 3 times, the chance of getting at least one "6" is 1 − (5/6)³ ≈ 42.1%. Use the "Trials" field in our Basic Probability tab to compute this.

Yes — for P(A AND B), our calculator assumes A and B are independent events, so P(A∩B) = P(A) × P(B). For P(A OR B), we use the inclusion-exclusion principle: P(A∪B) = P(A) + P(B) − P(A∩B). For dependent events, you would need the conditional probability P(A|B), which may differ from P(A). Always verify independence before relying on these results.

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. The formula is P(A|B) = P(A∩B) / P(B). For independent events, P(A|B) = P(A) — knowing B happened doesn't change A's likelihood. Our Multi-Event tab computes P(A|B) assuming independence unless you adjust the inputs accordingly.

Combinatorial numbers grow extremely fast — for example, 100C50 is approximately 1.0089 × 10²⁹, far beyond trillions. Our calculator displays these large values in scientific notation (e.g., 1.01e+29) to keep them readable. For practical purposes, values up to about 10¹⁵ are shown in full integer form. This is standard behavior for scientific and engineering calculators.

Two events are mutually exclusive if they cannot both occur at the same time. For mutually exclusive events, P(A∩B) = 0, so P(A∪B) = P(A) + P(B). A classic example: rolling a die and getting both a 1 and a 6 simultaneously is impossible — they're mutually exclusive. In contrast, drawing a Heart and drawing a King from a deck are not mutually exclusive (the King of Hearts exists).

Our calculator uses standard IEEE 754 double-precision floating-point arithmetic, providing accuracy to approximately 15-17 significant decimal digits. For combinatorics, we use efficient iterative multiplication algorithms to minimize rounding errors. All results are suitable for educational, professional, and betting purposes. For extremely high-precision needs (beyond 15 digits), specialized arbitrary-precision software may be required.

Probability and combinatorics are essential in: sports betting (converting odds, calculating implied probabilities), poker & card games (hand odds, outs), risk assessment (insurance, finance), quality control (defect rates in manufacturing), clinical trials (drug efficacy), lottery analysis, genetics (Punnett squares), and machine learning (Bayesian inference). Our tool covers the foundational calculations for all these domains.