Probability of 3 Events Calculator
Enter the probability of each of three independent events, as a decimal from 0 to 1 or as a percent, to get the chance all three occur, at least one, none, or exactly one.
Example: with P(A) — decimal 0-1 or a percent 0.5 · P(B) 0.5 · P(C) 0.5 → All three (A and B and C): 0.125 (12.5%).
- At least one (A or B or C)0.875 (87.5%)
- None occur0.125 (12.5%)
- Exactly one0.375 (37.5%)
Computed by the calculator below using its default values. Change any input to see your own numbers.
For independent events: AND multiplies, and 'at least one' is 1 minus the chance of none.
AND multiplies, OR adds up carefully
For independent events, the chance that all three happen is the product of their probabilities: P(A)·P(B)·P(C). Independence means one event's outcome does not shift another's odds. The chance that none happen is the product of the complements, (1-A)(1-B)(1-C), and the chance that at least one happens is simply one minus that. Building at least one from the none case avoids the double-counting that trips people up when they try to add overlapping events.
Exactly one, and the independence caveat
Exactly one event occurring means one happens while the other two do not, summed over the three ways to pick the winner: A(1-B)(1-C) + (1-A)B(1-C) + (1-A)(1-B)C. Every formula here assumes independence. If the events overlap or influence each other, use the general addition rule and conditional probabilities instead, because the simple products no longer hold.
How it’s calculated
For independent events with probabilities a, b, c: all three = a·b·c; none = (1-a)(1-b)(1-c); at least one = 1 - (1-a)(1-b)(1-c); exactly one = a(1-b)(1-c) + (1-a)b(1-c) + (1-a)(1-b)c. Enter each probability as a decimal from 0 to 1, or as a percent such as 50.
All formulas assume the three events are independent. For dependent or mutually exclusive events these products are wrong; use conditional probability or the addition rule.
Three independent events A, B, C
| You want | Formula |
|---|---|
| All three | P(A) x P(B) x P(C) |
| None | (1-A)(1-B)(1-C) |
| At least one | 1 - (1-A)(1-B)(1-C) |
| Exactly one | A(1-B)(1-C) + (1-A)B(1-C) + (1-A)(1-B)C |
Independence assumed: one event's outcome does not change another event's probability.
Common mistakes
- Adding probabilities for at least one as A + B + C. That overcounts the overlaps and can exceed 1; use 1 - (1-A)(1-B)(1-C).
- Assuming independence when events are linked. Drawing without replacement or shared causes break the simple product rule.
- Confusing at least one with exactly one. At least one includes the cases where two or all three events occur.
Frequently asked questions
How do you find the probability of 3 events?
For independent events, multiply for all three (A·B·C) and use 1 - (1-A)(1-B)(1-C) for at least one. Enter each probability as a decimal or percent and the calculator returns all, none, at least one, and exactly one.
What does independence mean here?
It means one event happening does not change the probability of another. Coin flips and separate dice are independent; cards drawn without replacement are not.
How is at least one different from exactly one?
At least one covers every case where one, two, or all three events occur. Exactly one counts only the cases where a single event happens and the other two do not.
Can I enter percentages?
Yes. A value between 1 and 100 is read as a percent, so 50 becomes 0.5. Values from 0 to 1 are treated as probabilities directly.
What if my events are mutually exclusive?
Then they cannot happen together, so all three is 0 and at least one is A + B + C when that sum is at most 1. This tool assumes independence, not exclusivity, so use the addition rule for that case.