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Probability Calculator for Two Events

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Probability Calculator (Two Events)

This probability calculator works out every combination of two events A and B. Enter P(A) and P(B) and say how the events are related: independent (one does not affect the other), mutually exclusive (they cannot both happen) or overlapping with a known P(A and B). The calculator returns the intersection, the union, both complements, exactly one, neither, and the two conditional probabilities.

The union uses the addition rule P(A or B) = P(A) + P(B) − P(A and B), which avoids counting the overlap twice. For independent events the overlap is P(A) × P(B); for mutually exclusive events it is 0. Conditional probability P(A | B) = P(A and B) ÷ P(B) is the chance of A once you know B happened. When the events are independent it equals P(A).

If you enter P(A and B) yourself, the calculator checks it is possible: it cannot exceed the smaller of P(A) and P(B), or be less than P(A) + P(B) − 1. For repeated independent trials, such as several coin tosses, use the binomial distribution calculator, and for counting outcomes see the combination calculator.

Result and step-by-step solution

P(A or B)
0.7

P(A and B) = 0.2; P(neither) = 0.3

  1. Intersection
    Independent: P(A and B) = P(A) × P(B) = 0.5 × 0.4 = 0.2
  2. Union (addition rule)
    P(A or B) = P(A) + P(B) − P(A and B) = 0.5 + 0.4 − 0.2 = 0.7
  3. Complements
    P(A’) = 1 − 0.5 = 0.5; P(B’) = 1 − 0.4 = 0.6
  4. Conditional probability
    P(A | B) = P(A and B) ÷ P(B) = 0.5; P(B | A) = 0.4
EventProbability
A and B0.2
A or B0.7
Exactly one of A, B0.5
Neither A nor B0.3
Not A0.5
Not B0.6
A given B0.5
B given A0.4

Frequently asked questions

How do I calculate the probability of A and B?
If A and B are independent, multiply: P(A and B) = P(A) × P(B). Otherwise use P(A and B) = P(A) × P(B | A).
How do I calculate the probability of A or B?
P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events the last term is 0, so the probabilities simply add.
What is the probability of neither event?
1 − P(A or B). With P(A) = 0.5 and P(B) = 0.4 independent, that is 1 − 0.7 = 0.3.
Are mutually exclusive events independent?
No (unless one has probability 0). If A happens, B cannot, so knowing A changes the probability of B to 0.

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Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.

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