Probability Calculator

Probability is the branch of mathematics that measures how likely an event is to happen, expressed as a number between 0 (impossible) and 1 (certain). It can equally be read as a percentage from 0% to 100%. This calculator works in three modes: basic probabilityof a single event, combinations and permutations for counting outcomes, and multiple events for combining two probabilities with the AND / OR rules. It is built for students, teachers, data and analytics learners, and anyone reasoning about chance — from dice and cards to risk, quality control, and everyday decisions.

Basic probability

For an experiment where every outcome is equally likely, the probability of an event A is P(A) = favourable outcomes / total outcomes. The result always lies between 0 and 1. For example, rolling a single fair six-sided die and asking for an even number gives 3 favourable outcomes (2, 4, 6) out of 6 total, so P = 3/6 = 0.5 = 50%. The probability that A does not happen is the complement: P(not A) = 1 − P(A). Probabilities are sometimes restated as odds, the ratio of favourable to unfavourable outcomes — a probability of 1/4 corresponds to odds of 1:3 in favour (or 3:1 against).

Combinations vs permutations

Many probability problems first require counting how many outcomes exist. Use a permutation when order matters and a combination when it does not, both built on the factorial n! (the product of every whole number up to n; by definition 0! = 1).

Combinations C(n, r): order does not matter. C(n,r) = n! / (r!(n−r)!) — for example choosing 3 people from 10 to form a team.

Permutations P(n, r): order matters. P(n,r) = n! / (n−r)! — for example arranging 3 of 10 people into distinct 1st / 2nd / 3rd places.

The two are linked by P(n,r) = C(n,r) × r!, because each chosen group can be ordered in r! ways.

Multiple events: the AND and OR rules

Two events are independent when one happening does not change the probability of the other (for example two separate dice rolls). For independent events the multiplication rule gives the chance of both: P(A and B) = P(A) × P(B). The general addition rule for the chance of at least one is P(A or B) = P(A) + P(B) − P(A and B); we subtract the overlap so it is not counted twice. Two events are mutually exclusive when they cannot both occur (for example rolling a 1 or a 6 on one die). Then P(A and B) = 0, and the addition rule simplifies to P(A or B) = P(A) + P(B).

Worked example

Suppose P(A) = 0.6 and P(B) = 0.4 for two independent events. Then P(A and B) = 0.6 × 0.4 = 0.24 (24%), and P(A or B) = 0.6 + 0.4 − 0.24 = 0.76 (76%). The chance that neither occurs is (1 − 0.6) × (1 − 0.4) = 0.4 × 0.6 = 0.24 (24%), which correctly matches 1 − P(A or B) = 1 − 0.76 = 0.24. Seeing the same answer two ways is a good check that the rules were applied correctly.

Tips

When a question asks for the probability of "at least one", it is almost always easier to use the complement: P(at least one) = 1 − P(none). Always confirm whether outcomes are equally likely before using favourable/total, and decide whether your two events are independent or mutually exclusive before picking a rule.

Common mistakes

A probability can never be below 0 or above 1 — if your arithmetic produces 1.2, an event was double counted (forgetting to subtract P(A and B) in the OR rule). Do not assume events are mutually exclusive just because they feel separate; mutually exclusive means they truly cannot co-occur. And do not multiply probabilities of events that are not independent — the simple product rule only holds when one event has no effect on the other.