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Three Independent Events: All, Any, One or None

Combine three mutually independent event probabilities to find all three, at least one, exactly one and no-event chances.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For three independent events each with a 50% chance, all three occur 12.5% of the time, at least one 87.5%, exactly one 37.5%, and none 12.5%.

All = ABC; none = (1 − A)(1 − B)(1 − C); any = 1 − none

Worked example

Enter these known values and leave the other values blank.

Probability of event A
50 %
Probability of event B
50 %
Probability of event C
50 %
Probability all three occur
12.5 %
Probability at least one occurs
87.5 %
Probability exactly one occurs
37.5 %
Probability none occur
12.5 %

Assumptions and limitations

  • The three events must be mutually independent. Pairwise independence alone is insufficient for the product formulas.
  • Each probability is from 0% to 100%, or zero to one when the decimal selector is chosen. Exactly-one probability sums three disjoint outcomes.
  • The tool does not calculate correlations, conditional probabilities or an exactly-two output. Some inverse combinations at boundaries cannot identify a unique missing probability.

Common questions

Can I use this when the events affect each other?

No. Dependence requires joint or conditional probabilities that are not inputs to this model.

Does at least one include all three?

Yes. At least one includes exactly one, exactly two and all three. Exactly one excludes the latter two cases.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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