Probability of 3 Independent Events Calculator
Combine three independent event probabilities and see the chance that all, at least one, exactly one, or none occur.
Probability of events
Enter a probability from 0 to 100. A trailing % is accepted.
Enter a probability from 0 to 100. A trailing % is accepted.
Enter a probability from 0 to 100. A trailing % is accepted.
Probability of...
All three occurring
6.00%
Intersection A ∩ B ∩ C
At least one occurring
82.00%
Union A ∪ B ∪ C
Exactly one occurring
45.00%
One success, two non-occurrences
None of them occurring
18.00%
Complement of at least one
Calculation details
| Outcome | Formula | Probability |
|---|---|---|
| All three occurring | P(A) × P(B) × P(C) | 6.00% |
| At least one occurring | 1 – (1 – P(A))(1 – P(B))(1 – P(C)) | 82.00% |
| Exactly one occurring | P(A)(1 – P(B))(1 – P(C)) + (1 – P(A))P(B)(1 – P(C)) + (1 – P(A))(1 – P(B))P(C) | 45.00% |
| None of them occurring | (1 – P(A))(1 – P(B))(1 – P(C)) | 18.00% |
How to use this probability of 3 events calculator
What this calculator does
This calculator combines the probabilities of three mutually independent events, labeled A, B, and C. From the three individual probabilities it computes four exact identities under that independence assumption: the chance that all three occur, at least one occurs, exactly one occurs, and none occur. It is useful when the occurrence of one event does not change the likelihood of either of the other two. It does not test independence from data and it does not handle dependence automatically; if the events influence one another, you need conditional or joint probabilities instead. The NIST probability tutorial's section on independence explains the underlying product rule and why the assumption matters.
When to use it
Use the calculator for independent repeated mechanisms or separate processes: for example, three independent component checks, three independent promotional responses, or three independently generated random outcomes. It is also useful for classroom exercises when you want to compare an intersection, a union, an exact-count event, and a complement using the same three inputs. Do not use it for mutually exclusive events from the same single trial, or for situations where event B becomes more or less likely after learning that A occurred.
How to calculate
- The calculator opens with a complete demonstration: P(A) = 60%, P(B) = 40%, and P(C) = 25%. The results and a validated Excel workbook are ready immediately.
- Replace the three probabilities with your own values. Choose Probability format as Percent for values from 0 to 100 or Decimal for values from 0 to 1. Switching format converts the current valid values rather than only changing the label.
- Read the four live outputs and the Calculation details table. Select Download Excel to export the current inputs, formulas, and results. Reset clears the demonstration inputs and results; because the cleared state is incomplete, Excel export remains unavailable until all three valid probabilities are entered again.
Input guide
P(A), P(B), and P(C) are required numeric probabilities. In Percent mode, enter 0 through 100 using a decimal point when needed, such as 60, 12.5, or 60%; a trailing percent sign is accepted. In Decimal mode, enter 0 through 1, such as 0.6 or 0.125. Commas and scientific notation are intentionally rejected so that an entry such as 1,5 is never silently reinterpreted. Increasing any one probability raises All three occurring and At least one occurring, but its effect on Exactly one occurring can be non-monotonic because more probability can shift mass toward two or three events occurring. A common mistake is entering 60 while Decimal mode is selected; that is outside the valid 0 – 1 range.
Probability format is a required display control, not a mathematical assumption. Percent and Decimal are two representations of the same canonical probability. For example, 25% and 0.25 are identical. Changing the format updates inputs, results, table values, and workbook number formatting while preserving the underlying probability.
Output guide
All three occurring is P(A ∩ B ∩ C), the exact probability that A, B, and C all happen. At least one occurring is P(A ∪ B ∪ C), so it includes cases with one, two, or all three events. Exactly one occurring counts only the three mutually exclusive cases in which one event happens and the other two do not. None of them occurring is the complement of at least one, and therefore those two outputs always add to 100% (or 1 in Decimal mode). A zero result means the corresponding outcome is impossible under the supplied probabilities; a value of 100% means it is certain under the model. The Calculation details table repeats these four output labels, shows the formula used for each row, and reports the same probability in the selected format.
Worked example
For the opening example, P(A) = 0.60, P(B) = 0.40, and P(C) = 0.25. Independence gives All three occurring = 0.60 × 0.40 × 0.25 = 0.06, or 6.00%. The probability of none is (1 – 0.60)(1 – 0.40)(1 – 0.25) = 0.18, or 18.00%, so At least one occurring is 1 – 0.18 = 82.00%. For exactly one event, the three disjoint cases total 0.60×0.60×0.75 + 0.40×0.40×0.75 + 0.40×0.60×0.25 = 0.45, or 45.00%. These are the same values shown on first open and written to the initial Excel workbook.
Learn more
Probability values lie between 0 and 1, with percentage notation providing the equivalent 0% to 100% scale; the MathWorld overview of probability summarizes those basic identities. For the union formula behind “at least one,” the inclusion – exclusion principle gives the general set relationship that prevents overlaps from being double counted.
How the formulas work
Independence lets joint probabilities factor into products. That is why the probability that all three events occur is simply P(A)P(B)P(C). Complements provide the shortest route to the probability that at least one event occurs: first calculate the probability that none occur, (1 – P(A))(1 – P(B))(1 – P(C)), then subtract that value from 1. This complement form is algebraically equivalent to applying inclusion – exclusion to three independent events.
Exactly one event requires three separate cases: A alone, B alone, or C alone. Those cases cannot happen simultaneously, so their probabilities can be added. The formula is P(A)(1 – P(B))(1 – P(C)) + (1 – P(A))P(B)(1 – P(C)) + (1 – P(A))(1 – P(B))P(C). Each term pairs one event probability with the complements of the other two.
Independence is the critical assumption
Two events are independent when knowing whether one occurred does not change the probability of the other. For three events, mutual independence is stronger than checking only each pair: the joint behavior must also factor consistently. This calculator assumes that condition is already justified. If you are modeling events that share a cause, occur in sequence with feedback, compete for the same resource, or arise from the same finite draw without replacement, the inputs may not be independent and the displayed products can be misleading.
A useful diagnostic is conceptual rather than numerical: ask whether learning the outcome of A would make you revise P(B) or P(C). If yes, use a conditional-probability model. Do not force dependent events into this calculator merely because three marginal probabilities are available.
Common interpretation mistakes
- At least one does not mean exactly one. It includes one, two, or all three events.
- All three and at least one are not disjoint categories, so their displayed probabilities should not be added together.
- Percent and decimal notation must not be mixed. In Percent mode, 25 means 25%; in Decimal mode, enter 0.25.
- Mutually exclusive events are generally not independent unless one has probability zero. If one event occurring prevents another from occurring, multiplication of marginal probabilities is not the right model.