Probability calculator
Probability of an event, its complement and of two independent events.
Fill in the fields and the result will appear here automatically.
Computes basic probability four ways: the share of favourable outcomes, the probability of the complement, and both the joint occurrence and the "at least one" case for two independent events. That last one is where intuition fails most often: "at least one" of two 50% events is not 100% but 75%, because these independent events overlap — the correct formula subtracts the overlap.
How it works
Formula and logic
The probability of an event is favourable outcomes divided by all outcomes. The complement is 1 − p. Both independent events: p₁ · p₂. At least one: p₁ + p₂ − p₁ · p₂. Favourable outcomes divided by total outcomes applies only when all outcomes are equally likely. Probabilities do add for disjoint events because P(A∩B) = 0; disjointness is different from independence. Counts are integers from 0 to 9007199254740991, with a positive total. Fields belonging to other modes are ignored.
Example
One favourable outcome out of six gives a probability of 0.1667, that is 16.667 %.
Fields and units
- What to compute — list option
- Favourable outcomes — unitless
- Total outcomes — unitless
- Favourable outcomes — unitless
- Total outcomes — unitless
- Probability of the first event — unitless
- Probability of the second event — unitless
- Probability of the first event — unitless
- Probability of the second event — unitless
How to use
- — Choose what you are computing.
- — Enter the outcomes or the event probabilities.
- — Read the probability as a fraction, a percentage and odds.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: outcome-count ratios for equiprobable models and fair dice OpenStax: independent intersection, union and distinction from disjoint events
- Limitation
- Favourable outcomes divided by total outcomes applies only when all outcomes are equally likely. Probabilities do add for disjoint events because P(A∩B) = 0; disjointness is different from independence.
FAQ
Why is "at least one" of two 50% events not 100%?
For two independent events, the overlap has probability 0.5×0.5 = 0.25. Adding 0.5+0.5 counts it twice, so P(A∪B) = 1−0.25 = 0.75. Two disjoint 50% events really would add to 100%, but they describe a different model.
What does "independent events" mean?
That the outcome of one does not affect the other: two coin tosses are independent, whereas drawing two cards without replacement is not, and needs a different formula.
How do I read the odds?
Odds of "5 to 1" mean five unfavourable outcomes for every favourable one. It is the same information as the probability, written differently.
Can a probability exceed one?
No. One means a certain event and nothing exceeds it — which is why there cannot be more favourable outcomes than outcomes in total.