Statistics

Probability Calculator (P(A∪B))

What this does

Combine two event probabilities into P(A or B), handling mutually exclusive and independent events correctly.

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Calculator inputs

How to use it

Using the probability calculator (p(a∪b))

  1. 01

    Enter both probabilities

    As percentages between 0 and 100.

  2. 02

    Declare the relationship

    Exclusive if both happening is impossible; independent if neither influences the other.

  3. 03

    Read P(A or B)

    The union; the chance that at least one of the events occurs.

Good to know

The double-counting trap

Naively adding probabilities works only for exclusive events. Rain tomorrow (40%) and rain the day after (40%) do not make 80%; independence subtracts their 16% overlap for 64%. Most real-world addition mistakes come from ignoring overlaps like this.

Exclusive versus independent

These are easy to confuse and almost opposite: mutually exclusive events are strongly dependent; one happening forces the other not to. Genuine independence is rare in everyday systems; reach for it only with a reason.

  • Cards from one draw: exclusive.
  • Two dice: independent.
  • Anything causal between them: neither mode fits.

How it's calculated

The math behind this calculator

Exclusive: P(A∪B) = P(A) + P(B) Independent: P(A∪B) = P(A) + P(B) − P(A)·P(B)

Mutually exclusive events (drawing a heart or drawing a club) can never co-occur, so probabilities add directly, clamped at 100% to catch over-specified inputs.

Independent events (two unrelated coin flips, two machines failing for separate reasons) overlap by chance, so we subtract the product P(A)·P(B) once to avoid double-counting the shared region.

Assumptions & limitations

  • Each probability entered as a percentage between 0 and 100.
  • Only exclusive and independent relationships are offered; dependent events need P(A∩B).
  • Clamping caps exclusive unions at 100% when inputs exceed feasibility.

Worked example

If a server has a 30% chance of a network fault and an independent 20% chance of a disk fault, the chance of at least one fault is 44%, not 50%; the double-counted 6% overlap is removed.

FAQ

Frequently asked questions

What about events that are neither exclusive nor independent?
Then you need P(A∩B) directly: P(A∪B) = P(A) + P(B) − P(A∩B). Enter the joint probability yourself rather than forcing a mode.
Why clamp at 100%?
Probabilities above 100% are impossible; exceeding it means the exclusive assumption contradicts your inputs somewhere upstream.
Can I get P(A and B)?
In the independent mode yes; it appears as a detail row equal to P(A)·P(B). Exclusive joint probability is zero by definition.
Do the inputs have to be percentages?
Enter them as percents (30, not 0.3); outputs are returned in the same percentage form.

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