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Probability Calculator for Two Events

Enter two event probabilities and their known intersection to calculate the union and complementary regions.

ABA only—A and B—B only—A or B—Neither—

Concept diagram; circle area is not proportional to probability.

Enter the event probabilities

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Decimal from 0 to 1.

Decimal from 0 to 1.

Must fit the overlap allowed by P(A) and P(B).

Two-event result

Enter P(A), P(B), and P(A∩B), then calculate.

How to use the two-event probability calculator

Use this page when P(A), P(B), and the probability that both occur are known. Values are decimals: 25% is entered as 0.25.

  1. Enter P(A) and P(B).
  2. Enter the known overlap P(A∩B); do not multiply P(A) and P(B) unless independence is established elsewhere.
  3. Select Calculate to see the union, complements, exclusive regions, and neither event.
  4. Check both the percentage and decimal forms in the result.
  5. Edit any input to invalidate the old result, or choose Clear to remove all inputs.

Addition and complement rules

The union is P(A∪B)=P(A)+P(B)−P(A∩B). Subtracting the intersection avoids counting outcomes that belong to both events twice. A complement is P(Aᶜ)=1−P(A), and the probability of neither event is 1−P(A∪B).

Why the overlap has limits

An intersection cannot exceed either event, so its upper limit is min(P(A),P(B)). It also cannot be smaller than max(0,P(A)+P(B)−1); otherwise the stated union would exceed 1. The calculator rejects combinations outside this feasible interval.

Independent and mutually exclusive are different

Independent events satisfy P(A∩B)=P(A)P(B), but this page never makes that assumption. Mutually exclusive events have intersection 0. Two events with positive probability cannot be both independent and mutually exclusive.

What the outputs mean

  • A only is P(A)−P(A∩B).
  • B only is P(B)−P(A∩B).
  • Neither is outside the entire union.
  • Percentages are rounded for display while calculations use the entered numeric values.

Worked example

With P(A)=0.60, P(B)=0.50, and P(A∩B)=0.20, the union is 0.90. A only is 0.40, B only is 0.30, and neither is 0.10.

Frequently asked questions

Does the calculator assume A and B are independent?

No. You enter the intersection directly, so dependent and independent cases can both be represented.

Can the intersection be larger than P(A)?

No. Every outcome in A∩B is also in A and in B.

Should I enter 25 or 0.25 for 25%?

Enter 0.25. Every input uses the inclusive range from 0 to 1.

Why was my overlap rejected?

The three probabilities would not describe a possible pair of events. Check the displayed feasible interval rule.