Probability Calculator - Single, Complement, And, Or

What is probability?

Probability measures how likely an event is to occur, expressed as a number from 0 (impossible) to 1 (certain). A probability of 0.5 means a 50% chance. This calculator handles the four most common probability operations taught in statistics and probability courses.

Single event probability

Enter P(A) directly as a decimal. For example, rolling a 6 on a fair die has probability 1/6 ≈ 0.1667. The calculator displays the result as a fraction, decimal, and percentage.

Complement: P(A′)

The complement of event A is the probability that A does not occur: P(A′) = 1 − P(A). If there is a 25% chance of success, there is a 75% chance of failure.

P(A or B): Union

The union is the probability that at least one of A or B occurs. For mutually exclusive events (they cannot both occur at the same time), P(A ∪ B) = P(A) + P(B); their probabilities must add to no more than 1. For independent events, the overlap is P(A)·P(B), so the union is P(A) + P(B) − P(A)·P(B). Events that merely overlap are not automatically independent: enter the known overlap P(A ∩ B) and the calculator uses P(A) + P(B) − P(A ∩ B) without making that assumption.

P(A and B): Intersection

The intersection is the probability that both A and B occur. For independent events: P(A ∩ B) = P(A) × P(B). For dependent events, enter P(B|A): the conditional probability of B given A: P(A ∩ B) = P(A) × P(B|A).

The relationship selector is part of the calculation, not just a label. It makes the selected assumption visible in the displayed formula. For example, if P(A) = 0.6, P(B) = 0.5, and the known overlap is 0.2, P(A or B) is 0.6 + 0.5 − 0.2 = 0.9. Treating those same marginal probabilities as independent would instead give 0.8, which illustrates why the overlap matters.

Bayes' theorem

Bayes' theorem updates the probability of an event given new evidence:

P(A|B) = P(B|A) × P(A) ÷ P(B)

Classic example: a medical test for a rare disease (prevalence 1%) has 99% sensitivity and 99% specificity. If you test positive, the probability you actually have the disease is only ~50% — not 99%. This base rate neglect fallacy is why Bayes’ theorem matters for interpreting test results and statistical evidence.

Common probability mistakes

• Gambler’s fallacy: believing that past independent outcomes affect future ones. A coin that has landed heads 10 times in a row still has a 50% chance of heads on the next flip — the coin has no memory.
• Prosecutor’s fallacy: confusing P(evidence | innocent) with P(innocent | evidence). A 1-in-a-million DNA match probability does not mean the defendant has only a 1-in-a-million chance of being innocent — base rates matter.

Worked examples