Union with overlap
If P(A) = 0.50, P(B) = 0.40, and P(A ∩ B) = 0.20, then P(A ∪ B) = 0.50 + 0.40 − 0.20 = 0.70, or 70%.
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Probability formulas
| Calculation | Formula | Use it when |
|---|---|---|
| Union | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | You need the chance that A, B, or both occur. |
| Independent intersection | P(A ∩ B) = P(A) × P(B) | Knowing A occurred does not change the probability of B. |
| General intersection | P(A ∩ B) = P(A | B) × P(B) | A conditional probability is known; independence is not required. |
| Conditional probability | P(A | B) = P(A ∩ B) ÷ P(B) | You restrict the sample space to outcomes where B occurred; P(B) must be positive. |
| Complement | P(Ac) = 1 − P(A) | You need the chance that A does not occur. |
Notation: ∪ means “or,” ∩ means “and,” the vertical bar in P(A | B) means “given,” and Ac means “not A.” “Or” is inclusive here: it includes outcomes where both events happen.
How union, intersection, and conditional probability fit together
For two events, the intersection is the overlap: outcomes in both A and B. The union covers every outcome in A or B, including that overlap. Adding P(A) and P(B) counts the overlap twice, which is why the union formula subtracts P(A ∩ B) once.
A conditional probability changes the reference group. P(A | B) asks: among outcomes where B happened, what fraction also belong to A? This produces the division rule P(A | B) = P(A ∩ B) / P(B). Rearranging gives the general multiplication rule for an intersection.
Independence is an assumption, not a default
Events A and B are independent only when learning that one occurred does not change the probability of the other. Equivalent tests include P(A | B) = P(A), when P(B) > 0, and P(A ∩ B) = P(A)P(B). Do not multiply marginal probabilities unless independence is stated or justified.
Consistency limits for two events
A valid intersection must satisfy max(0, P(A) + P(B) − 1) ≤ P(A ∩ B) ≤ min(P(A), P(B)). These Fréchet bounds ensure the overlap is neither too small to fit both events inside the whole sample space nor larger than either event. This calculator checks those limits.
Worked probability examples
Independent intersection
If a fair coin lands heads with probability 0.50 and an independent die shows 6 with probability 1/6, both occur with probability 0.50 × 1/6 = 1/12, about 8.33%.
Conditional probability
If 20% of all outcomes are in both A and B and 40% are in B, then P(A | B) = 0.20 / 0.40 = 0.50, or 50%.
Complement
If a delivery arrives on time with probability 0.92, its probability of not arriving on time is 1 − 0.92 = 0.08, or 8%.
Frequently asked questions
How do I calculate the probability of A or B?
Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtract the intersection because it is included once in P(A) and once in P(B).
How do I calculate the probability of A and B?
If the events are independent, multiply P(A) by P(B). Otherwise use P(A ∩ B) = P(A | B)P(B), or the equivalent formula P(B | A)P(A).
How do I calculate conditional probability?
Divide the joint probability by the conditioning event: P(A | B) = P(A ∩ B) / P(B). The conditioning event must have probability greater than zero.
What is the complement rule?
A and not A cover every outcome without overlap, so P(not A) = 1 − P(A).
Are mutually exclusive events independent?
Not when both have positive probability. Mutually exclusive events have P(A ∩ B) = 0, but independent events would require P(A ∩ B) = P(A)P(B), a positive value in that case.
Can I enter percentages instead of decimals?
Yes. Choose Percent and enter values between 0 and 100. Changing the input format converts values already entered in the form.
Why are my probabilities inconsistent?
The overlap cannot exceed either event and cannot be less than max(0, P(A) + P(B) − 1). The calculator reports which limit is violated so you can check the source values or assumptions.
Are my inputs private?
Yes. All calculations happen locally in your browser. The calculator does not send or store the probability values you enter.