Probability Calculator

Compute union, intersection, complement and conditional probabilities of events.

About Probability Calculator

Probability quantifies how likely an event is on a scale from 0 (impossible) to 1 (certain). The addition rule gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive they cannot both occur, so P(A ∩ B) = 0. If they are independent, knowing B gives no information about A, so P(A ∩ B) = P(A) × P(B). Conditional probability P(A | B) = P(A ∩ B)/P(B) updates the likelihood of A given that B is known to have occurred.

Formulas

Addition rule

P(A or B) = P(A) + P(B) − P(A and B)

Independent events

P(A and B) = P(A) × P(B) when A and B are independent

Conditional probability

P(A given B) = P(A and B) / P(B)

Complement

P(not A) = 1 − P(A)

Worked example

With the default values below, Probability Calculator returns Probability of 0.760000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Calculateone of P(A ∪ B) — Union (A or B), P(A ∩ B) — Intersection (A and B), P(Aᶜ) — Complement (not A), P(A | B) — Conditional (A given B): P(A ∪ B) — Union (A or B)
P(A)accepts 0–1: 0.6
P(B)accepts 0–1: 0.4
Relationship between A and Bone of Independent, Mutually exclusive (disjoint), Neither — enter P(A ∩ B) directly: Independent
P(A ∩ B)accepts 0–1: 0.1

Results

Probability: 0.760000
P(A ∩ B) usedThe joint probability of A and B both occurring.: 0.240000

Step by step

1Addition rule
2P(A ∩ B)= 0.240000
3Substitute= 0.760000

Assumptions and limits

The result is derived only from the 5 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot happen simultaneously — if A occurs, B cannot (P(A ∩ B) = 0). Independent events can both occur, but knowing one happened gives no information about the other (P(A | B) = P(A)). These are opposite extremes: two events with P(A) > 0 and P(B) > 0 cannot be both mutually exclusive and independent.

When is the addition rule P(A ∪ B) = P(A) + P(B)?

Only when A and B are mutually exclusive, so P(A ∩ B) = 0. For any other pair of events you must subtract P(A ∩ B) to avoid double-counting outcomes where both events occur.

Can a conditional probability be greater than the original probability?

Yes. If A and B are positively correlated, learning that B occurred increases the probability of A. For example, the probability of a positive cancer test is higher given that the patient actually has cancer than in the general population.