Probability Calculator - Bayes, Conditional, And/Or

How to use this tool

1. Choose single event, complement, P(A or B), P(A and B), conditional P(A|B), or Bayes P(A|B).
2. Enter the relevant probabilities as decimals from 0 to 1; for example, 0.5 means 50%.
3. For OR, choose mutually exclusive, independent, or known overlap. For AND, choose independent or enter P(B|A).
4. For conditional probability, enter P(A ∩ B) and P(B). For Bayes, enter P(A), P(B|A), and either P(B) or P(B|A′).
5. Read the result as a fraction, decimal, and percentage, with the formula, event partition when available, and Bayes complement and evidence probability.

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 single events, complements, unions, intersections, conditional probabilities, and Bayes' theorem.

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.

All related probabilities

When the selected relationship supplies a justified overlap and both event probabilities are known, the calculator also shows the complete two-event partition: both events, either event, exactly one event, neither event, A only, B only, and the complements of A and B. For example, independent events with P(A) = 0.6 and P(B) = 0.5 have an intersection of 0.3 and a union of 0.8. A dependent AND calculation from P(B|A) gives the intersection, but without the marginal P(B) it cannot determine the rest of the partition.

Conditional probability: P(A|B)

Conditional probability asks how likely A is among outcomes where B has occurred. Enter the joint probability P(A ∩ B) and P(B); the calculator divides the joint by P(B). The denominator must be greater than zero, and the joint cannot exceed P(B). This direction matters: P(A|B) and P(B|A) usually have different values.

Bayes' theorem

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

P(A|B) = P(B|A) × P(A) ÷ P(B)

The Bayes mode can use a supplied total evidence probability P(B), or derive P(B) from the false-positive rate P(B|A′). For a rare-disease illustration, use a 1% prior, 99% sensitivity, and a 1% false-positive rate. A positive result then gives a posterior probability of about 50%, because positive results also occur among the much larger group without the disease. This is a base-rate example, not a diagnosis; actual tests and medical decisions require clinical context.

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