Probability Calculator

Background.

If there is a 50% chance of rain on Saturday and a 50% chance on Sunday, is a wet weekend certain? Adding the two says 100%, and that is wrong: the sum counts twice every weekend on which it rains both days. Take that overlap out and, if the two days were independent, the chance of at least one wet day is 0.5 + 0.5 − 0.25 = 75%. Weather isn't independent from one day to the next — a wet Saturday makes a wet Sunday more likely — so the real figure is usually lower still. Two small corrections, and between them they cover most of what goes wrong when people combine chances by instinct.

Four rules are on offer. A single probability can be restated as a percentage, a complement and odds. Two independent probabilities can be combined as both happening, P(A) × P(B), or at least one happening, P(A) + P(B) − P(A) × P(B). And the complement turns the chance of something into the chance of it not happening, 1 − P(A), which is often the quickest route to an awkward answer: the chance of at least one six in four rolls of a die is 1 − (5/6)⁴ = 0.5177.

Probabilities go in as decimals from 0 to 1, so a 30% chance is 0.3. The two-event rules are right only when the events are independent, meaning that learning one happened leaves the chance of the other unchanged: separate coins, separate dice, draws that are put back. Cards dealt from one deck, two symptoms of the same illness and the weather on neighbouring days are not independent, and multiplying their probabilities can give an answer that is badly wrong.

What is probability calculator?

A probability is a number from 0 to 1 measuring how likely an event is, with 0 meaning impossible and 1 certain. When all the outcomes of an experiment are equally likely, it is the number of favourable outcomes over the total, so a six on a fair die is 1/6. The equally-likely condition is easy to break without noticing. The totals of two dice run from 2 to 12, but they are not equally likely: a 7 can be made six ways out of 36, so its chance is 6/36, while a 2 needs double ones, 1/36. More generally, the rules come from three axioms Andrey Kolmogorov set out in 1933: no probability is negative, the probability that one of all the possible outcomes occurs is 1, and the probabilities of events that cannot happen together simply add. Every rule used here follows from those three plus one definition. Two events are independent when P(A and B) = P(A) × P(B), the formal way of saying that knowing one tells you nothing about the other. Mutually exclusive is a different idea: the events can't both happen, so P(A and B) = 0. Odds carry the same information as a probability in ratio form: the chance an event happens divided by the chance it doesn't, P ÷ (1 − P).

How to use this calculator.

1. 01Pick the rule in What do you want to compute?: Single event to restate one probability, Both events occur for P(A and B), At least one occurs for P(A or B), Complement for P(not A).
2. 02Enter Probability of A as a decimal: divide a percentage by 100, or work out a fraction first — 1/6 is 0.1667 and 4/52 is 0.0769. Values below 0 or above 1 are rejected. Rounding a fraction to four places costs almost nothing: 0.1667 × 0.1667 gives 0.0278, the same as 1/36 to four decimals.
3. 03For the two-event rules, enter Probability of B as well; Single event and Complement ignore it. Before trusting the result, check that one event does not make the other more or less likely.
4. 04Read the result as a decimal and as a percentage. The complement line is 1 minus the result, so in Complement mode it hands back P(A); the odds line is the result divided by its complement.
5. 05For three or more independent events, chain the steps: combine A and B, then enter that result as A and the next event as B. For at least one success over many identical tries, 1 − (1 − p)ⁿ is quicker.

The formula.

Single event: result = P(A). Both, for independent events: P(A and B) = P(A) × P(B). At least one, for independent events: P(A or B) = P(A) + P(B) − P(A) × P(B). Complement: P(not A) = 1 − P(A). Whatever result R the rule produces, the percentage is 100 × R, the complement is 1 − R and the odds in favour are R ÷ (1 − R).

The OR rule is the general addition rule, P(A) + P(B) − P(A and B), with independence supplying the overlap as P(A) × P(B). Without the subtraction the overlap is counted twice, which is how two 50% chances turn into a false certainty. For events that cannot happen together the overlap is zero and plain addition is correct, but this rule always subtracts P(A) × P(B), so it is the wrong tool for them. Rolling a 1 or a 6 with one die is 1/6 + 1/6 = 0.3333. Putting 1/6 and 1/6 through the OR rule gives 11/36 = 0.3056, which answers a different question: the chance of at least one six when two separate dice are rolled.

Dependent events need a conditional probability in the multiplication rule: P(A and B) = P(A) × P(B given A). Drawing two aces in a row from one deck is 4/52 × 3/51 = 0.0045, or 1 in 221, because the first ace leaves three in a smaller deck. Treating the draws as independent, 4/52 × 4/52, gives 0.0059, about 31% too high.

Odds and probability convert with P = odds ÷ (1 + odds) and odds = P ÷ (1 − P). At P = 1 the odds are infinite, and the result displays 1.000 × 10³⁰⁸ as a stand-in.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Kolmogorov, A.N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability). Ergebnisse der Mathematik. Springer, Berlin. The monograph that introduced the modern axiomatic foundation of probability theory.
2. [2]Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume 1, 3rd edition, Chapter I: 'The Sample Space'. John Wiley & Sons, New York. The canonical introductory treatment of sample spaces, the addition and multiplication rules, and independence.
3. [3]Casella, G. & Berger, R.L. (2002). Statistical Inference, 2nd edition, Chapter 1: 'Probability Theory'. Duxbury / Cengage, Pacific Grove, CA. The standard graduate textbook treatment of Kolmogorov's axioms, conditional probability, and Bayes' rule.
4. [4]Ross, S.M. (2019). A First Course in Probability, 10th edition. Pearson, New York. A widely-adopted undergraduate text covering independence, the inclusion-exclusion principle, and worked examples on coin flips, dice, and card draws.
5. [5]Laplace, P.-S. (1814). Essai philosophique sur les probabilités (A Philosophical Essay on Probabilities). Mme Ve Courcier, Paris. The first systematic non-technical exposition of probability, introducing the classical (equally-likely-outcomes) definition and the principle of insufficient reason.